Proving a language is not regular using the Myhill-Nerode Theorem
The words $0^n1$ are pairwise distinguishable, since $0^n10^n$ is a palindrome but $0^n10^m$ isn't (for $n \neq m$).

With a bit more work, you can show that in fact all words are pairwise distinguishable. See for example here or here.