(a) Das ist nichts, auf das man stolz sein sollte.
(b) Das ist nichts, auf was man stolz sein sollte.
(c) Das ist nichts, worauf man stolz sein sollte.
Which ones are acceptable, and what are the differences?
In colloquial language, you'll probably encounter all three (with no difference in meaning), but (c) is my guess for the canonical version.
In addition to the other answer, some further notes:
This is more generic and is not restricted to "nichts".
That said, there are still phrases where it would sound a little odd to use "auf was", for instance:
Though, admittedly, when rephrasing the sentence you, again, can use both variants:
Still, remember it's colloquialism and you would be well-advised to go with worauf when being uncertain.
Finally a note on whether to go with "auf das" or "auf was".
So, regarding the rule of thumb you would definitely say
In the sentence "Das ist nichts, ..." it sounds OK to use das, though. And indeed it has more Google hits. I'm not sure why you rather take the relative pronoun das in that case.