  In this paper we consider unramified coverings of the moduli space
$\mathcal{M}_g$ of smooth projective complex curves of genus $g$. Under some
hypothesis on the branch locus of the finite extended map to the
Deligne-Mumford compactification, we prove the vanishing of the vector space of
holomorphic 1-forms on the preimage of the smooth locus of $\mathcal{M}_g$.
This applies to several moduli spaces, as the moduli space of curves with
2-level structures, of spin curves and of Prym curves. In particular, we obtain
that there are no non-trivial holomorphic 1-forms on the smooth open set of the
Prym locus.
