  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 $\overline{\mathcal
R}_{g,G}$ 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 $\overline{\mathcal M}_g$ 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 $G=S_3$, the $J$-locus is
empty, which is the first fundamental step in evaluating the Kodaira dimension
of $\overline{\mathcal R}_{g,S_3}$.
