Is the moduli space of Prym curves (curves $C$ with square root of $\mathcal{O}_C$, compactified via admissible covers  by Beauville) of a given genus $g$ normal? Why?
There are various references. The one I like is the following, particularly Remark 1.3.3. MR2007376 (2005b:14049) Reviewed Abramovich, Dan(1BOST); Corti, Alessio(4CAMB); Vistoli, Angelo(IBOLO) Twisted bundles and admissible covers. (English summary) Special issue in honor of Steven L. Kleiman. Comm. Algebra 31 (2003), no. 8, 3547–3618. 14H10 (14A20 14H30) http://arxiv.org/pdf/math/0106211.pdf 


The stack $\overline{\mathcal{P}}_{g}$ of Prym curves is a smooth DeligneMumford stack. This implies that its coarse moduli space $\overline{P}_{g}$ is normal with at most fine quotient singularities. 

