What is known about $S$ if $\{\langle M\rangle : L(M)\in S \}$ is recursive or recursively enumerable

Question

For $L_S=\{\langle M\rangle : L(M)\in S \}$ what is known about $S$ in case of:

1. $L_S\in RE$

2. $L_S\in R$

Answer 1

Rice's theorem states that if $L_S$ is not trivial (i.e., is not $\varnothing$ nor all languages) then $L_S$ can't be decided (it might be computably enumerable, though).

An extension to Rice's theorem states that $L_S$ is computably enumerable if and only if after replacing $S$ with $S\cap RE$, all the following hold.

1. For all $L_1, L_2$ computably enumerable, if $L_1 \in S$ and $L_1 \subseteq L_2$, then $L_2 \in S$.

2. If $L \in S$, there is a finite subset $L' \subseteq L$ so that $L' \in S$.

3. The set of finite languages in $S$ is computably enumerable.

Proofs are given at here.

Answer 2

Take a look at Rice's theorem and its extensions.

Basically, Rice's theorem and its extension state:

1. If $\emptyset\neq S\subset RE$ then $L_S\notin R$
2. If $\emptyset \neq S\subset RE$ and $\Sigma^*\notin S$ (please fix me if this is wrong, this is what I remember) then $L_S\notin RE$