Japt, 6 5 bytes

Õmx1R

Try it
Run it twice (returns original string)

Zsh +coreutils, _{139 110 109 105 95} 55 bytes

for i ({1..`wc -L<a`})cut -c$i a|rs -eg0|sed 's/ *$//'

Try it online!

Assumes input file a has lines of varying length. Pads the lines with spaces Slices, reshapes, and trims trailing spaces.

`wc -L<a`               # get the longest line of 𝐚̲
for i ()cut -c$i a      # take vertical slices of 𝐚̲ at each $i
|rs -eg0|sed 's/ *$//'  # reshape and remove trailing spaces

_{Naïve solution (TIO): for i ({1..`wc -L<a`})cut -c $i a|tr -d \\n&&echo
95b 105b 109b 110b 139b}

K (ngn/k), 33 bytes

{"\n"/(|&\|^:)_'+(|/#'x)$x}@"\n"\

Try it online!

• {...}@"\n"\ split (implicit) input on newlines, passing into lambda
  • +(|/#'x)$x right-pad each string to maximum length with spaces, then transpose
  • (|&\|^:)_' drop trailing spaces from each line
  • "\n"/ join with newlines and (implicitly) return

Setanta, 191 190 bytes

gniomh(s){s=roinn@s("
")t:=0u:=""le i idir(0,fad@s)t=uas(t,fad(s[i]))le i idir(0,t){le j idir(0,fad@s)u+=(i<fad(s[j])&s[j][i])|" "nuair-a u!=""&u[-1]==" " u=cuid@u(0,fad@u-1)u+="
"}toradh u}

try-setanta.ie link

R, 106 bytes

for(j in Map(substr,list(s<-scan(,"",sep="\n")),n<-1:max(nchar(s)),n))cat(ifelse(j==""," ",j),"\n",sep="")

Attempt This Online!

I have chosen to write the complete program: it grabs the manual input and prints the result out.

The text is transformed to a list of characters vectors (puts together all the 1st letters, the 2nd etc.). This is done by mapping the substr function, and the list size corresponds to the longest textline. When there is not enough characters an empty character is produced (this is why I prefer to use substr instead of strsplit: there'll be no vector recycling!).

The program then iterates through this list and conditionally outputs the characters: instead of the empty one there will be a space, or the character itself otherwise; the newlines are added at the end of each vector.

Perl 5 + -plF -M5.10.0, 22 bytes

Utilises ANSI escape sequences and unprintables.

$"="..";$_=".[1;$.H@F"

Try it online!

Explanation

This code first changes the list separator ($") value to be a form-feed and a backspace, then we set the implicitly printed variable $_ to be a string that starts with the ANSI escape sequence for 'start printing at line 1 column $. (where $. is the current line of text: \x1b1;$.H) and interpolates the list @F (which is a list of all the characters on that line, populated by autosplit (-a, implied by -F) with an empty split pattern (-F)) which places the contents of $" in between each item, moving the cursor vertically down instead of continuing output after the previous character.

Uiua, 31 bytes

/$"_\n_"⍚⍜⇌(▽\↥≠@ .)⍉⬚@ ⊜∘≠@\n.

Try it in the pad!

Ruby: 88 characters

(Posted because it's shorter then the other Ruby solutions. Not checked whether my code introduces anything new compared to those. If you already posted a Ruby solution and you feel this is mostly a copy of yours, please comment and I will retire my answer.)

f=->t{l=t.split$/;r=[""]*m=l.map(&:size).max;l.map{|l|m.times{|i|r[i]+=l[i]||" "}};r*$/}

Attempt This Online!

Sample run:

irb(main):001:0> f=->t{l=t.split$/;r=[""]*m=l.map(&:size).max;l.map{|l|m.times{|i|r[i]+=l[i]||" "}};r*$/}
=> #<Proc:0x99a9e68@(irb):1 (lambda)>

irb(main):002:0> sample='I am a text.
irb(main):003:0' Transpose me.
irb(main):004:0' Can you do it?'
=> "I am a text.\nTranspose me.\nCan you do it?"

irb(main):005:0> puts f[sample]
ITC
 ra
aan
mn
 sy
apo
 ou
ts
eed
x o
tm
.ei
 .t
  ?
=> nil

irb(main):006:0> puts f[f[sample]]
I am a text.
Transpose me.
Can you do it?
=> nil

MATL, 5 bytes

!Z{Zv

Try it on MATL Online

Input is taken implicitly, and conveniently, the shorter lines get spaces added to them by the environment to make them all equal in length.

! - transpose

Z{ - Convert to cell array (since we'll need to hold arrays of different lengths after the next step)

Zv - deblank i.e. remove the trailing spaces

K (oK), 43 bytes

Solution:

{"\n"/{(-+/&\32=|x)_x}@'+(|/#:'x)$x:"\n"\x}

Try it online!

Explanation:

Break on newlines, pad lines to same lengths, transpose, trim trailing whitespace and join on newlines. Feels like there has to be an easier way...

{"\n"/{(-+/&\32=|x)_x}@'+(|/#:'x)$x:"\n"\x} / the solution
{                                         } / lambda
                                         x  / input x
                                    "\n"\   / split (\) on "\n"
                                  x:        / save as x
                                 $          / pad right with left
                         (      )           / do this together
                            #:'x            / count (#:) each (') x
                          |/                / max
                        +                   / transpose
      {              }@'                    / apply (@) lambda to each (')
                    x                       / input x
                   _                        / drop
       (          )                         / do this together
                |x                          / reverse (|) x
             32=                            / equal to ASCII 32 aka " "
           &\                               / min
         +/                                 / sum
        -                                   / negate
 "\n"/                                      / join (/) on "\n"

C (MinGW), 272 262 259 254 234 229 228 bytes

-15 bytes thanks to ceilingcat

The TiO link requires the string.h include, but MinGW does not.

n,a,i,j;*f(int*s){char*p,**L=a=n=i=0,*r;for(;p=strtok(L?0:s,"\n");j>a?a=j:0)j=strlen((L=realloc(L,++n*8))[n-1]=p);for(p=r=calloc(n+1,a+1);i<a;i++,*p++=10){for(j=0;j<n;j++)*p++=i<strlen(L[j])?L[j][i]:32;while(p[-1]==32)p--;}s=r;}

Try it online!

JavaScript, 102 100 bytes

q=>[...q].reduce((r,e,j)=>(y=w.map((e,i)=>w[i][j]||" ").join``.trimEnd())?r+y+`
`:r,"",w=q.split`
`)

Run it here:

f = q=>[...q].reduce((r,e,j)=>(y=w.map((e,i)=>w[i][j]||" ").join``.trimEnd())?r+y+`
`:r,"",w=q.split`
`);

console.log(f(`I am a text.
Transpose me.
Can you do it?`));

MATLAB: 24 42 29 bytes

@(s)deblank(string(char(s)'))

Usage

s=["I am a text.";"Transpose me";"Can you do it?"];
@(s)deblank(string(char(s)'))
ans(s)

Output

ans =

  14×3 char array

    'ITC'
    ' ra'
    'aan'
    'mn'
    ' sy'
    'apo'
    ' ou'
    'ts'
    'eed'
    'x o'
    'tm'
    '.ei'
    '  t'
    '  ?'

Might be cheating here, as it's hard (as far as I can see) to get a string with newlines into MATLAB without using an array, so I went ahead and used an array for the input. Other than that, it does what it says on the tin - turns the string into a matrix of chars, transposes it and trims the whitespace.

Edit: Changed (and lengthened) the program as I didn't see the requirement for whitespace to be trimmed.

Edit: -13 bytes thanks to sundar.

Octave, 35 29 bytes

@(s)strjoin(cellstr(s'),'\n')

Try it online!

(-1 byte thanks to Luis Mendo, -5 bytes thanks to strjoin from this SO answer)

05AB1E, 6 bytes

ζεðÜ}»

Try it online.
Try it when taking the output as input-list again.

Explanation:

ζ         # Zip the input-list, swapping rows and columns
 ε  }     # For-each:
  ðÜ      #  Remove trailing spaces
     »    # Join the resulting list by new-lines (and output implicitly)

Python 2, 87 101 bytes

lambda s:T('\n'.join(map(T,map(''.join,zip(*[t.ljust(len(s))for t in s.split('\n')])))))
T=str.rstrip

Try it online!

Lua, 203 189 bytes

t={{}}i=1m=0(...):gsub(".",function(c)n=#t[i]if c=="\n"then i=i+1t[i]={}else t[i][n+1]=c end m=m<=n and n+1or m end)
for x=1,m do for p=1,i do io.write(t[p][x]or" ")end _=m<=x or print()end

Try it online!

I saw another Lua solution here, but I don't think there's a problem with posting 2 solutions on the same language. If there is, tell me :)

Excel VBA, 154 bytes

Takes input from [A1] and outputs to the console.

x=Split([A1],vbLf):u=UBound(x):For i=0To u:l=Len(x(u)):m=IIf(m>l,m,l):Next:For i=1To m:For j=0To u:s=s+Mid(x(j)&String(m,32),i,1):Next:?RTrim(s):s="":Next

Ungolfed

x=Split([A1],vbLf)              ''  Split input by lines
u=UBound(x)                     ''  Get number of lines
For i=0To u                     ''  Iterate over lines
l=Len(x(u))                     ''  Get length of line `i`
m=IIf(m>l,m,l)                  ''  Compare to max length; Store Max
Next                            ''  Loop
For i=1To m                     ''  Iterate over chars in line
For j=0To u                     ''  Iterate over lines
s=s+Mid(x(j)&String(m,32),i,1)  ''  Append `i`th char of `j`th line to `s` 
Next                            ''  Loop
?RTrim(s)                       ''  Remove right whitespace, print
s=""                            ''  Reset `s`
Next                            ''  Loop

Pyth, 2 bytes

.t

Try it here!

Performs the built in transpose function on an input list, outputting a list of lines.

Pyth, 7 bytes

V.t.zdN

Try it here!

The above program, modified to use the input and output format specified in the challenge.

Retina, 40 bytes

P`.+
~L0$`.
L,$.%'vs,$.(x$%=),`.+
m` *$

Try it online!

Based on this program that I created to transpose any size of rectangle, which uses Leo's suggestion for transposing a rectangle of a known size.

Edit: Golfed more and no longer has trailing spaces on lines.

Javascript, 103

s=>[...s].map((_,i)=>s.split`
`.map(b=>r+=b[q=b[i]||q,i]||' ',r=q='')&&r.replace(/ *$/,q?`
`:q)).join``

Less golfed

s=>[...s].map(
     // we need 'i' ranging from 0 to the length of the longest input line
     // so we scan all the input string, that is surely longer
     // but we need to check that after some point the output must be empty
     (_, i) => ( 
       r = '', // the current output row, starts empty
       q = '', // flag to check if we are beyond the longest line
       s.split('\n') // split in rows
       .map( 
         b => ( // for each input row in b
           q = b[i] || q, // if there is a char at position i in b, i goes to q
           r += b[i] || ' ' // add to output the char at position i or a fill space
         )
       ),
       q // if q is still '', we are beyond the longest input line 
       ? r.replace(/ *$/,`\n`) // trim leading space and add newline
       : '' // no output 
     )
   ).join('')

Test

F=
s=>[...s].map((_,i)=>s.split`
`.map(b=>r+=b[q=b[i]||q,i]||' ',r=q='')&&r.replace(/ *$/,q?`
`:q)).join``

function go() {
  var text=I.value
  var output = F(text)
  O.textContent = output
}

go()

#I { width:50%; height:5em }

<textarea id=I>I am a text.
Transpose me.
Can you do it?</textarea><br>
<button onclick='go()'>Transpose</button>
<pre id=O></pre>

Lua, 205 bytes

L={}C=0 R=io.read I=R()while I~=""do L[C]=I C=C+1 I=R()end M=0 for i=0,C-1,1 do if#L[i]>M then M=#L[i]end end for i=1,M,1 do for j=0,C-1,1 do io.write(L[j]:sub(i,i)==""and" "or L[j]:sub(i,i))end print()end

Try it online!

Ungolfed

lines={}
count=0
R=io.read
input=R()

-- reads input in until an empty line is found
-- lines are added to the lines table
while input~="" do
    lines[count]=input
    count=count+1;
    input=R()
end

-- find the length of the longest string
max_len=0
for i=0,count-1,1 do
    if #lines[i]>max_len then
        max_len=#lines[i]
    end
end

-- loop through each character of each string
for i=1,max_len,1 do
    for j=0,count-1,1 do
        -- print the string's next character
        -- or a space if there is not another character
        if lines[j]:sub(i,i)=="" then--this if statement is golfed into a
            io.write(" ")            --pseudo-ternary statement in the golfed version
        else
            io.write(lines[j]:sub(i,i))
        end
    end
    print()
end

J, 28 26 Bytes

Saved 2 bytes thanks to frownyfrog

t=.,@:(,&LF"1)@|:@:>@cutLF

Takes a string, returns a string. I'm not sure if there's a shorter version of the 'cutopen' function verb that I could use.

There's also the shorter

t=.|:@:>@cutLF

But I'm not sure it falls within the OP's guidelines, as it returns an array of characters.

How it works:

                     cutLF   | Splits the input on new lines and boxes them
                    @        | Composes verbs (as does @:, but they're not equal)
                   >         | Unboxes this, forming an array of the lines
                 @:          |
               |:            | Transposes the array
      (      )@              |
       ,&LF                  | Appends a new line...
           "1                | To each row of the array
    @:                       |
   ,                         | Flatten the result
t=.                          | Assign this verb to t

The other version works the same, but doesn't convert the transposed array to a properly formatted string.

Examples:

NB. Define a multi-line string

    text =: 0 : 0
I am a text.
Transpose me.
Can you do it?
)

    t text
ITC
 ra
aan
mn    NB. There's whitespace after the 'n' here, but I assume it doesn't count as trailing since it's part of the original string
 sy
apo
 ou
ts 
eed
x o
tm 
.ei
 .t
  ?

    t t text
I am a text.     NB. Again, whitespace here, but it's part of the argument of the second 't' (added by the first 't' to keep columns straight)
Transpose me. 
Can you do it?

Perl 5, 81 bytes

chomp(@a=<>);say((join"",map{(substr$_,0,1,"")||$"}@a)=~s/\s+$//r)while"@a"=~/\S/

Try it online!

Pyth, 11 bytes

# aYcw1;.tY

Try it online!

Takes input as a string, outputs a list of lists

Pyth, 25 bytes

# aYcw1=+Z1;jbcj"".n.tY)Z

Takes input as a string, outputs a string.

Try it online!

PHP, 95+0 108+1 110 bytes

foreach($input as $s)for($y=0;~$c=$s[$y];)$r[$y++].=$c>"
"?$c:" ";echo rtrim(join("
",array_map(rtrim,$r)));

takes input from STDIN; breaks at the first empty line. Run with -nr or try it online.

Still trying to find something shorter than the additional rtrim to get rid of the second newline.

Bash, 124 bytes

D=`mktemp -d`;split -l1 - $D/;for F in $D/*;do grep -o . $F>$F+
done;paste $D/*+|sed -e's/\([^\t]\)\t/\1/g;s/\t/ /g;s/ *$//'

It reads standard input and writes standard output. Try it:

echo $'I am a text.\nTranspose me.\nCan you do it?' | script.sh

How it works:

• split input into single lines (files in temporary directory $D)
• split lines into single characters using grep (files *+)
• layout characters side-by-side using paste (TAB-separated columns)
• remove alignment TABs, replace filler TABs with BLANKs, trim using sed

Edit:

• -9: Removed tidy-up code ;rm -r $D (thanks Tim)
• -2: use + instead of _ as suffix and shorten ${F}_ to $F+
• -3: remove prefix L from split result files

JavaScript, 177 173 bytes

s=>{l=s.split`
`.sort((a,b)=>b.length-a.length)[0];for(i=0,t=[];i<l.length;i++){t[i]=s.split`
`.map(i=>i+" ".repeat(l.length-i.length)).map(j=>j[i]).join``}return t.join`
`}

s=>{l=s.split`\n`.sort((a,b)=>b.length-a.length)[0];for(i=0,t=[];i<l.length;i++){t[i]=s.split`\n`.map(i=>i+" ".repeat(l.length-i.length)).map(j=>j[i]).join``;}return t.join`\n`}

If you can help me shorten this up, please do so.

Groovy, 98 chars

{i->o=[].withDefault{''};i.readLines().each{it.toList().eachWithIndex{c,d->o[d]+=c}};o.join('\n')}

online

ungolfed:

{i->
o=[].withDefault{''};//create list with empty string as default value 
i.readLines()
.each{
    it.toList() //split every line to characters
    .eachWithIndex{ 
        c,d->o[d]+=c //append to string from list with right index
    }
};
o.join('\n')}//join list with newlines
}

CJam, 32 25 bytes

CJam is newer than this challenge, so this answer is not eligible for being accepted.

Considerably shortened by user23013.

qN/_z,f{Se]}z{S+e`);e~N}%

Test it here.

qN/                       "Read input, split into lines.";
   _z,                    "Transpose, get length (find maximum line length).";
      f{Se]}              "Pad each line to that length with spaces.";
            z             "Transpose.";
             {         }% "Map this block onto each line in the result.";
              S+          "Add a space to ensure there's at least one.";
                e`        "Run-length encode.";
                  );      "Discard the trailing run of spaces.";
                    e~    "Run-length decode";
                      N   "Push a newline.";

K, 56

This should meet the spec now.

Accepts a string, returns a string.

{`/:{$[" "=*|x;|(+/&\" "=|x)_|x;x]}'x@'/:!max@#:'x:`\:x}

.

k)f:{`/:{$[" "=*|x;|(+/&\" "=|x)_|x;x]}'x@'/:!max@#:'x:`\:x}
k)f"I am a text.\nTranspose me.\nCan you do it?"
"ITC\n ra\naan\nmn\n sy\napo\n ou\nts\need\nx o\ntm\n.ei\n .t\n  ?\n"
k)f f"I am a text.\nTranspose me.\nCan you do it?"
"I am a text.\nTranspose me.\nCan you do it?\n"

Perl (92+1)

reads stdin and writes to stdout. adding 1 to the score for say

@L=map[grep!/\n/,split//],<>;do{$_=join'',map shift@$_||$",@L;s/ +$//;say}while grep@$_>0,@L

Bash+coreutils+sed, 83

eval paste `sed 's/.*/<(fold -w1<<<"&")/'`|expand -t2|sed 's/\(.\) /\1/g;s/ \+$//'

fold and paste do the important work. The rest is just formatting.

Accepts input from stdin and outputs to stdout:

$ < tr.txt ./transposefile.sh
ITC
 ra
aan
mn
 sy
apo
 ou
ts
eed
x o
tm
.ei
 .t
  ?
$ < tr.txt ./transposefile.sh | ./transposefile.sh
I am a text.
Transpose me.?
Can you do it
$ 

Haskell: 191 chars

import Data.List
h [] = ' '
h (x:_) = x
t [] = []
t ([]:x) = t (" ":x)
t ((x:y):z) = takeWhile (any (/= ' ')).map (dropWhileEnd (== ' '))$(x:map h z):t (y:[s |(_:s)<-z])
m = (unlines.t.lines)

Mathematica, 95 chars

f=""<>Riffle[Thread@PadRight@Characters@StringSplit[#,"\n"]//.{0->" ",{x___," "..}:>{x}},"\n"]&

Python 2.7 (97 79 94 90)

EDIT: Missed the function requirement;

I'm fairly sure this will be improved on since I'm sort of a beginner here, but to start with;

c=lambda a:'\n'.join(''.join(y or' 'for y in x).rstrip()for x in map(None,*a.split('\n')))

The code uses a simple split to split the string into a vector of rows. It then uses map with a function value as None (the unity function) and the splat operator to transpose and pad the vector (similar functionality to zip_longest in Python3)

The rest of the code just maps None to space, trims and reassembles the matrix to a single string again.

>>> a = 'I am a text.\nTranspose me.\nCan you do it?'
>>> c(a)                                                                            
'ITC\n ra\naan\nmn\n sy\napo\n ou\nts\need\nx o\ntm\n.ei\n .t\n  ?'
>>> len("""c=lambda a:'\n'.join(''.join(y or' 'for y in x).rstrip()for x in map(None,*a.split('\n')))""")
88
# (+2 since `\n` is considered by `len` to be a single char)

PHP 227 217

Here's my attempt:

function f($s){$p=array_map;$s=$p(str_split,explode("
",$s));foreach($s as&$l)$l=array_pad($l,max($p(count,$s)),' ');array_unshift($s,null);foreach(call_user_func_array($p,$s)as$l)$r.=rtrim(implode($l))."
";return$r;}

Ungolfed:

<?php
function f($s) {
    $s=array_map(str_split, explode("\n",$s));
    foreach($s as &$l)
        $l=array_pad($l, max(array_map(count, $s)),' ');
    array_unshift($s, null);
    foreach(call_user_func_array(array_map, $s)as$l)
        $r.=rtrim(implode($l))."\n";
    return $r;
    }
$s="I am a text.
Transpose me.
Can you do it?";
echo f($s);
echo strlen(f($s));

Python 89 103 chars

def f(a):return'\n'.join([''.join(i).rstrip()for i in zip(*[j+' '*99 for j in a.split('\n')])]).rstrip()

I feel dirty. 90 104 chars for industrial strength version. :^)

R, 171

function(e){p=strsplit
x=t(plyr::rbind.fill.matrix(lapply(p(p(e,"\n")[[1]],""),t)))
x[is.na(x)]=" "
cat(apply(x,1,function(y)sub(" *$","",paste(y,collapse=""))),sep="\n")}

Usage example:

text <- "I am a text.
Transpose me.
Can you do it?"


(function(e){p=strsplit
x=t(plyr::rbind.fill.matrix(lapply(p(p(e,"\n")[[1]],""),t)))
x[is.na(x)]=" "
cat(apply(x,1,function(y)sub(" *$","",paste(y,collapse=""))),sep="\n")})(text)

ITC
 ra
aan
mn
 sy
apo
 ou
ts
eed
x o
tm
.ei
 .t
  ?

Trailing whitespace is removed.

MATHEMATICA 117 chars

t = "I am a text.\nTranspose me.\nCan you do it?";

f=(m=Length/@(f=Flatten[Characters/@StringSplit[#,"\n"],{{2},{1}}])//Max;
StringJoin@@@(PadLeft[#,m," "]&/@f)//Column)&

AutoHotkey 210

f(i){
StringSplit,o,i,`n
m:=0
loop % o0 {
a:=A_index
if (t:=Strlen(p:=o%a%))>m
m:=t
StringSplit,l%a%,o%a%
}
loop % m {
a:=A_index,n:=""
loop % o0
n.=(j:=l%A_index%%a%)=""?" ":j
s.=Rtrim(n," ") "`n"
}
return s
}

Test

text=
(
I am a text.
Transpose me.
Can you do it?
)
msgbox % f(text)

Haskell

import Data.List
main = interact (unlines . transpose . lines)

It was so short, I needed to add in white space...

Scheme/Racket 113

The text:

(define t (list 
    (string->list "I am a text.") 
    (string->list "Transpose me.")
    (string->list "Can you do it?")
))

Without new lines and extra white spaces:

(define s(λ(v x)(if(= x 0)'()(cons(list->string(car v))(s(cdr v)(- x 1))))))(s(apply map list t)(length(car t)))

The user-friendly version

(define text (list 
    (string->list "I am a text.") 
    (string->list "Transpose me.")
    (string->list "Can you do it?")
))

(define transpose
    (λ(text length)
        (if (= length 0)
            '()
            (cons (list->string (car text)) (transpose (cdr text) (- length 1)))
)))

(transpose (apply map list text) (length (car text)))

Ruby 111

Golfed:

def f t;s=t.lines;s.map{|l|l.chomp.ljust(s.map(&:size).max).chars}.transpose.map{|l|l.join.rstrip+?\n}.join;end

Ungolfed:

def transpose_text(text)
  max_length = text.lines.map(&:size).max
  text.lines.map do |line|
    line.chomp.ljust(max_length).chars
  end.transpose.map do |chars|
    chars.join.rstrip + "\n"
  end.join
end

Ruby has an array transpose function, so this simply pads the lines out, turns them into an array of characters, uses Ruby's Array#transpose function, then turns the array of characters back into lines.

Golfing it was simply using single-character identifiers, removing spaces, using a temporary for text.lines, and putting the calculation for max_length inline (there are no points for efficiency).

PHP 194

function x($a){$a.="\n";$s=strlen($a);$i=0;while($c<$s)if($a{$c}!="\n")$b[$i++].=$a{$c++};else{$c++;for(;$i<$s;$i++)$b[$i].=" ";$i=0;}ksort($b);return rtrim(implode("\n",array_map("trim",$b)));}

Non-golfed:

function x($a) {
    $a.="\n";
    $s=strlen($a);
    $i=0;
    while($c<$s)
        if($a{$c}!="\n")
            $b[$i++].=$a{$c++};
        else{
            $c++;
            for(;$i<$s;$i++)
                $b[$i].=" ";$i=0;
        }
    ksort($b);
    return rtrim(implode("\n",array_map("trim",$b)));
}

This is my first golfing attempt, so please be kind! Also, tips/suggestions would be greatly appreciated!

Ruby — 144 characters

Here's my first attempt, golfed:

def f t
t.split(?\n).each{|l|l<<' 'until l.size==t.split(?\n).map(&:size).max}.map{|x|x.split('')}.transpose.map{|l|l.join.rstrip}.join(?/n)
end

For output, run puts f text where text is any multi-line string adhering to the rules above. The ungolfed version is below:

def text_transpose(text)
  lines = text.split(?\n)
  maxlen = lines.map(&:size).max
  lines.each { |line| line << ' ' until line.size == maxlen }
       .map  { |line| line.split('') }.transpose
       .map  { |char| char.join.rstrip }.join(?\n)
end

For a similar, but ultimately better solution in Ruby, check out Wayne Conrad's code above.

GolfScript, 51 chars

n%.{,}%$-1=" "*:y;{y+y,<}%zip{n\+0{;).32=}do}%((;\+

This is a first attempt; I suspect it can be improved. Most of the code is to comply with the padding and trailing space removal requirements — without them, just n%zip n* would suffice.

Ps. The following 46-character version will do the job for the given sample input, but will crash if any input column consists entirely of spaces:

n%.{,}%$-1=" "*:y;{y+y,<}%zip{0{;).32=}do]}%n*

I assume that's enough to disqualify it, even if the challenge doesn't explicitly say so.

Python 2.7 - 115 chars:

oneliner:

>>> a
'I am a text.\nTranspose me.\nCan you do it?'

>>> "".join(["".join(i)+'\n' for i in zip(*[x+" "*(len(max(a.splitlines(), key=len))-len(x)) for x in a.splitlines()])])
'ITC\n ra\naan\nmn \n sy\napo\n ou\nts \need\nx o\ntm \n.ei\n .t\n  ?\n'

and in a cleaner printing:

>>> print "".join(["".join(i)+'\n' for i in zip(*[x+" "*(len(max(a.splitlines(), key=len))-len(x)) for x in a.splitlines()])])
ITC
 ra
aan
mn 
 sy
apo
 ou
ts 
eed
x o
tm 
.ei
 .t
  ?

in 115 chars:

>>> len(""""".join(["".join(i)+'\n' for i in zip(*[x+" "*(len(max(a.splitlines(), key=len))-len(x)) for x in a.splitlines()])])""")
115

C++ (243 characters)

Here's a function that takes and returns a string.

I could've shaved a couple dozen chars, but decided to keep it as not-stupid code (runs fast, reads okay). Maybe I only decided to do that because this is my first code golf... I'm not hardcore enough yet :)

string f(string s){stringstream ss(s);vector<string> v;for(size_t i=0;getline(ss,s);++i){if(v.size() < s.size())v.resize(s.size());for(size_t j=0;j<s.size();++j){v[j].resize(i,' ');v[j].push_back(s[j]);}}s="";for(auto& i:v)s+=i+'\n';return s;}

With formatting:

string f(string s)
{
    stringstream ss(s);
    vector<string> v;

    for(size_t i = 0; getline(ss, s); ++i)
    {
        if(v.size() < s.size())
            v.resize(s.size());

        for(size_t j = 0; j < s.size(); ++j)
        {
            v[j].resize(i, ' ');
            v[j].push_back(s[j]);
        }
    }

    s = "";
    for(auto& i : v)
        s += i + '\n';

    return s;
}

C (278 bytes)

Edit: This actually breaks the rules, since it takes a filename in as an argument but writes to stdout. I'll edit it later to write to a file and then print the filename to stdout.

This is my first code golf ever, so have mercy. Some plain old C. Place the input in test.txt and let it run!

clang transpose.c -o transpose && ./transpose test.txt

#import <stdio.h>
#import <stdlib.h>
#import <string.h>

#define BUFFER_SIZE 1024

#define MAX(A,B) ((A)>(B)?(A):(B))

int main(int argc, char **argv) {
    char line[BUFFER_SIZE];

    FILE *f; int nLines, maxLen;

    f = fopen(argv[1], "r");
    while(!feof(f) && fgets(line, BUFFER_SIZE, f)) {
        nLines++;
        maxLen = MAX(maxLen, strlen(line));
    }
    fclose(f);

    for (int charPos = 0; charPos < maxLen; charPos++) {
        f = fopen(argv[1], "r");
        for (int linePos = 0; linePos < nLines; linePos++) {
            fgets(line, BUFFER_SIZE, f);
            printf("%c", charPos < strlen(line) && line[charPos] != '\xA' ? line[charPos] : ' ');
        }
        printf("\n");
        fclose(f);
    }

    return 0;
}

By using short variable names, removing gratuitous formatting, and allowing file handles to leak, and disabling all warnings this is reduced to 278 bytes. (Since this uses implicit imports, it may not link properly on all systems. Works on my machine!)

#import <stdio.h>
int main(int C,char**V){char L[1024];int A,B,D,I,J,*F=fopen(V[1],"r");while(!feof(F)&&fgets(L,1024,F)){A++;D=strlen(L);B=B>D?B:D;}for(I=0;I<B;I++){F=fopen(V[1],"r");for(J=0;J<A;J++)fgets(L,1024,F)&&printf("%c",I<strlen(L)&&L[I]!='\n'?L[I]:' ');printf("\n");}}

J (31 40)

f=:3 :';(,&LF@dlb&.|.)&.><"1|:>LF cut y'

This is a function that takes a string, and returns a string (i.e. a character vector with linefeeds inserted in the right places, and not a matrix.)

Edit: no trailing whitespace on any line.

Test:

   f=:3 :';(,&LF@dlb&.|.)&.><"1|:>LF cut y'

   string=:stdin''
I am a text.
Transpose me.
Can you do it?
^D

   $string
42
   $f string
53
   f string
ITC
 ra
aan
mn
 sy
apo
 ou
ts
eed
x o
tm
.ei
 .t
  ?