Convert a date to xkcd notation

JavaScript (ES6), 168 173

As an anonymous function. Using template strings, there is a newline near the end that is significant and included in the byte count.

d=>d.replace(/\d/g,c=>m=(l=((o[c]=o[c]||[b,c,b])[p/2&~1]+=++p).length)>m?l:m,t=[m=p=b='',b,b],o=[])&o.map(x=>x&&x.map((x,i)=>t[i]+=(x+'     ').slice(0,m+1)))||t.join`
`

Less golfed

f=d=>(
  // get the indices in o and the columns width in m
  m=0,
  p=0,
  o=[],
  d.replace(/\d/g,c=>(
    o[c] = o[c]||['',c,''], // for each found digit :array with top indices, digit, bottom indices
    o[c][p/2 & ~1] += ++p, // (p/2 and not 1) maps 0..3 to 0, 4..7 to 2
    l = o[c].length,
    m = l>m ? l : m // max indices string length in m 
  )),
  // build the output in t
  t=['','',''],
  o.map(x=> x && x.map(
    (x,i) => t[i]+=(x+'     ').slice(0,m+1)) // left justify, max value of m is 4
  ),
  t.join`\n` // return output as a newline separated string
)   

Test snippet

f=d=>
  d.replace(/\d/g,c=>m=(l=((o[c]=o[c]||[b,c,b])[p/2&~1]+=++p).length)>m?l:m,t=[m=p=b='',b,b],o=[])&
  o.map(x=>x&&x.map((x,i)=>t[i]+=(x+'     ').slice(0,m+1)))
  ||t.join`\n`


console.log=x=>O.textContent+=x+'\n'

;['2013-02-27','2015-12-24','2222-11-11','1878-02-08','2061-02-22','3564-10-28','1111-11-11']
.forEach(t=>console.log(t+'\n'+f(t)+'\n'))

<pre id=O></pre>

Ruby, 200 195 189 178 162 157 characters

(156 characters code + 1 character command line option)

o={}
i=0
$_.gsub(/\d/){o[$&]||=['','']
o[$&][i/4]+="#{i+=1}"}
o=o.sort.map &:flatten
puts [1,0,2].map{|i|o.map{|c|c[i].ljust o.flatten.map(&:size).max}*' '}

Sample run:

bash-4.3$ ruby -n xkcd-date.rb <<< '2013-02-27'
2  3  1  4    
0  1  2  3  7 
5     67    8 

bash-4.3$ ruby -n xkcd-date.rb <<< '2222-11-11'
     1234
1    2   
5678     

bash-4.3$ ruby -n xkcd-date.rb <<< '3564-10-28'
      1 4 2 3  
0 1 2 3 4 5 6 8
6 5 7         8

Python 2.7, 308 310 bytes

i=raw_input().replace("-","")
s,w=sorted(set(i)),len
x,m={},0
for c in s:
    q,v=i,[""]*2
    while c in q:a=str(-~q.index(c)+(w(i)-w(q)));v[int(a)>4]+=a;q=q[q.index(c)+1:]
    m,x[c]=max(m,max(map(w,v))),v
for l in[0,1]:print"".join((lambda x:x+(-~m-w(x))*" ")("".join(x[n][l]))for n in s)+"\n"+(" "*m).join(s)*(-l+1)

Wow, fixing it only costed 2 bytes!

The date doesn't have to be separated, the date can be any length, it doesn't have to be a date, it could be any string (but dashes are removed). The middle part looks pretty golfable to me.