Let $\overline{\mathcal{M}}_g$ be the moduli stack of genus $g$ stable nodal curves and let $\overline{M}_g$ denote its coarse moduli space. In 1969, in the paper "The irreducibility of the space of curves of given genus", Deligne and Mumford constructed and showed that $\overline{\mathcal{M}}_g$ is a smooth Deligne-Mumford stack, and proved that $\overline{M}_g$ is an irreducible projective variety. Although $\overline{\mathcal{M}}_g$ is smooth, $\overline{M}_g$ may be singular.
When is $\overline{M}_g$ singular?
The simplest case is $\overline{M}_0$, which is a point and hence smooth by definition. My question is: for which $g$ is $\overline{M}_g$ singular?