Shortest program to split a string at non-digits without RegExps
APL, 13 chars

(or 28 / 30 bytes, read below)

{⍵⊂⍨⍵∊∊⍕¨⍳10}


I see you've banned GolfScript from your question. I understand your sentiment, but I hope this community won't eventually ban APL, because it's a truly remarkable programming language with a long history, not to mention a lot of fun to code in. Maybe it could just be scored differently, if people feel it's competing unfairly. I'll post my thoughts on this matter to that thread you've linked.

On that same token, I've always added a footnote to my APL posts, claiming that APL could be scored as 1 char = 1 byte. My claim rests on the fact that a few (mostly commercial) APL implementations still support their own legacy single-byte encoding, with the APL symbols mapped to the upper 128 byte values. But maybe this is too much of a stretch, in which case you may want to score this entry as 28 bytes in UTF-16 or 30 bytes in UTF-8.

Explanation

{        ⍳10}  make an array of naturals from 1 to 10
       ⍕¨      convert each number into a string
      ∊        concatenate the strings into one (it doesn't matter that there are two 1s)
    ⍵∊         test which chars from the argument are contained in the digit string
 ⍵⊂⍨           use it to perform a partitioned enclose, which splits the string as needed


Examples

      {⍵⊂⍨⍵∊∊⍕¨⍳10} 'ab5c0x'
 5  0 
      {⍵⊂⍨⍵∊∊⍕¨⍳10}  'z526ks4f.;8]\p'
 526  4  8 


The default output format for an array of strings does not make it clear how many strings are there in the array, nor how many blanks. But a quick manipulation to add quotes should make it clear enough:

      {q,⍵,q←'"'}¨ {⍵⊂⍨⍵∊∊⍕¨⍳10} 'ab5c0x'
 "5"  "0" 
      {q,⍵,q←'"'}¨ {⍵⊂⍨⍵∊∊⍕¨⍳10}  'z526ks4f.;8]\p'
 "526"  "4"  "8"