Criteria for cuspidal S_k singularities and their applications

We give useful criteria for S_1 singularities in the Mond classification table, and cuspidal S_k singularities. As applications, we give a simple proof of a result given by Mond and a characterization of cuspidal S_k singularities for the composition of a cuspidal edge and a fold map indicated by Arnol'd for the case k=1.