  We further analyze the moduli space of stable curves with level structure
provided by Chiodo and Farkas in \cite{AA}. Their result builds upon Harris and
Mumford analysis of the locus of singularities of the moduli space of curves
and shows in particular that for levels 2, 3, 4, and 6 the locus of
noncanonical singularities is completely analogous to the locus described by
Harris and Mumford, it has codimension 2 and arises from the involution of
elliptic tails carrying a trivial level structure. For the remaining levels (5,
7, and beyond), the picture also involves components of higher codimension. We
show that there exists a component of codimension 3 for levels $\ell=5$ and
$\ell\geqslant 7$ with the only exception of level 12. We also show that there
exists a component of codimension 4 for $\ell=12$.
