Is there a way a proving a language regular/non-regular that works for every possible language? In my theory of computing class, we've been talking about how to prove languages regular and non-regular. I've heard of methods like the pumping lemma and Kolmogorov complexity to prove languages non-regular, while proving languages regular revolves around building a DFA, or using the Myhill-Nerode theorem. All of these approaches fail for some language. Is there some approach that will work for all regular/non-regular languages?
No, there isn't. It is undecidable, even for context-free languages. It is decidable for deterministic context-free languages.

References are easy to find with Google; for instance,


Undecidable Problems for Context-free Grammars, by Hendrik Jan Hoogeboom
Deciding whether a context-free language is regular [closed], on the Theoretical Computer Science Stack Exchange, in which user babou lists


A regularity test for pushdown machines,  R.E. Stearns, Information and Control, 1967 (full text by clicking on the page)
Regularity and Related Problems for Deterministic Pushdown Automata, Leslie G. Valiant, JACM 1975