Decidability of code properties

We explore the borderline between decidability and undecidability of the following question: “Let C be a class of codes. Given a machine ${\mathfrak{M}}$ of type X, is it decidable whether the language $L({{\mathfrak{M}}})$ lies in C or not?” for codes in general, ω-codes, codes of finite and bounded deciphering delay, prefix, suffix and bi(pre)fix codes, and for finite automata equipped with different versions of push-down stores and counters.