Are recursively enumerable languages closed under shuffle?
Hint: A language is recursively enumerable if and only if it can be recognised by a nondeterministic Turing machine. Use this fact to build a nondeterministic recogniser for $\textsf{shuffle}(L_1,L_2)$ from the deterministic recognisers for $L_1$ and $L_2$. What is it, that one most use nondeterminism to guess?