subsets of non regular language I know that there are many languages that are context free but not regular
like $\{a^n b^n :n>0\}.$
But I want to know if every context free but non-regular language has infinitely many non-regular subsets?
Thank you. 

Yes.
If $L$ is context free but non-regular, then for every positive integer n, let $L_n\subseteq L$ be a language such that $|L-L_n|=n$. If $L_n$ were regular, then $L$ would be regular, since an automaton which decides $L_n$ could be (non-deterministically for simplicity) expanded to decide $L_n$ and exactly all the finite strings removed from $L$, and thus the automaton also decides $L$. Thus we conclude that none of $L_n$ may be regular.
