Machines whose languages are their own encoding
Using the recursion theorem, for any Turing machine $T$ you can construct a Turing machine $M$ such that on input $\langle M \rangle$, $M$ executes $T$ (on the empty tape), and otherwise $M$ immediately rejects. You can use this to show that $S$ is not decidable. Using the same ideas you can explore its recognizability and co-recognizability.