  We describe the singular locus of the compactification of the moduli space
$R_{g,l}$ of curves of genus $g$ paired with an $l$-torsion point in their
Jacobian. Generalising previous work for $l\le 2$, we also describe the
sublocus of noncanonical singularities for any positive integer $l$. For $g\ge
4$ and $l=3,4, 6$, this allows us to provide a lifting result on pluricanonical
forms playing an essential role in the computation of the Kodaira dimension of
$R_{g,l}$: for those values of $l$, every pluricanonical form on the smooth
locus of the moduli space extends to a desingularisation of the compactified
moduli space.

Algebraic Geometry