Abstract: 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 l-torsion line bundle. They showed that for l ≤ 6 and different from 5 pluricanonical forms extend over any desingularization. This opens the way to a computation of the Kodaira dimension without desingularizing, as done by Farkas and Ludwig for l = 2, and by Chiodo, Eisenbud, Farkas and Schreyer for l = 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^l is isomorphic to a chosen power of the canonical bundle. New loci of canonical and non-canonical singularities appear and 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 = S3 we approach the evaluation of the Kodaira dimension of the moduli space, and present the remaining obstacles to compute it.