Singularities of the moduli space of level curves

We describe the singular locus of the compactification of the moduli space R-g,R-l of curves of genus g paired with an l-torsion point in their Jacobian. Generalising previous work for l <= 2, we also describe the sublocus of noncanonical singularities for any positive integer l. For g >= 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,R-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.