In a recent paper Chiodo and Farkas described the singular locus and the
locus of non-canonical singularities of the moduli space of level curves. In
this work we generalize their results to the moduli space Rg,G of curves with a G-cover for any finite group G. We show that
non-canonical singularities are of two types: T-curves, that is singularities
lifted from the moduli space Mg of stable curves, and
J-curves, that is new singularities entirely characterized by the dual graph
of the cover. Finally, we prove that in the case G=S3, the J-locus is
empty, which is the first fundamental step in evaluating the Kodaira dimension
of Rg,S3.Comment: 35 pages. arXiv admin note: text overlap with arXiv:1504.0056