In a series of recent papers, Chiodo, Farkas and Ludwig carried out a deep analysis of the singular locus of the moduli space of stable (twisted) curves with an $\ell$-torsion line bundle. This opens the way to a computation of the Kodaira dimension without desingularizing, as done by Farkas and Ludwig for $\ell=2$, and by Chiodo, Eisenbud, Farkas and Schreyer for $\ell=3$.

We can generalize this works in two directions. At first we treat roots of line bundles on the universal curve systematically: we consider the moduli space of curves C with a line bundle $L$ such that $L^{\xx\ell}\cong\omega_C^{\xx k}$.

New loci of canonical and non-canonical singularities appear for any $k\not\in\ell\Z$ and $\ell>2$, we provide a set of combinatorial tools allowing us
to completely describe the singular locus in terms of dual graphs. Furthermore, we treat moduli spaces of curves with a $G$-cover where $G$ is any finite group.
In particular for $G=S_3$ we can evaluate the Kodaira dimension of the moduli space.