Can we detect perfect matchings in P? in NP? in coNP?
Edmonds' "blossom algorithm" determines membership in MATCH deterministically in time $O(|E|\,|V|^2)$.

You answer this kind of question by searching for algorithms that solve your problem as efficiently as possible and, if there don't seem to be any, searching for NP-completeness results. If you don't find either, then you need to come up with your own algorithm or your own hardness proof.