Is the set of surjective recursive functions in RE/coRE?
Rice-Shapiro proves that $L$ is not RE nor coRE.

Let $F$ be the set of computable surjective functions. $L$ is the the index set associated to $F$.

Assume by contradiction $L$ is RE. By Rice-Shapiro (compactness), since the identity function belongs to $F$, some finite restriction of the identity must belong to $L$ as well. But a finite-domain function can not be surjective -- contradiction.

Assume by contradiction $\bar L$ is RE. The always undefined function $g$ belongs to $\bar F$, so by Rice-Shapiro (monotonicity), $\bar F$ contains any computable extension of $g$, including the identity function, which is surjective. Contradiction.