  In a series of recent papers, Chiodo, Farkas and Ludwig carry out a deep
analysis of the singular locus of the moduli space of stable (twisted) curves
with an $\ell$-torsion line bundle. They show that for $\ell\leq 6$ and
$\ell\neq 5$ pluricanonical forms extend over any desingularization. This
allows to compute the Kodaira dimension without desingularizing, as done by
Farkas and Ludwig for $\ell=2$, and by Chiodo, Eisenbud, Farkas and Schreyer
for $\ell=3$. Here 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 $L^{\otimes\ell}\cong\omega_C^{\otimes k}$. New loci of canonical
and non-canonical singularities appear for any $k\not\in\ell\mathbb{Z}$ and
$\ell>2$, we provide a set of combinatorial tools allowing us to completely
describe the singular locus in terms of dual graph. We characterize the locus
of non-canonical singularities, and for small values of $\ell$ we give an
explicit description.
