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 have finite quotient singularity.


**When is $\overline{M}\_g$ smooth?**


The simplest case is $\overline{M}\_0$, which is a point and hence smooth by definition. My question is then equivalent to the following: for which $g>0$ is $\overline{M}\_g$ smooth? Is it never smooth?


**Edit 1:**  

I forgot to mention the base scheme is $\text{Spec }\mathbb{C}$.



When is the coarse moduli space of genus $g$ stable curves smooth?
A more detailed description of the singular locus of $\mathrm{M}\_g$ is as follows.


**Theorem.** Let $\mathrm{C}$ be a smooth curve of genus $g$. 


If $g=2$, then $[\mathrm{C}]$ is a singular point of $\mathrm{M}\_2$ if and only if $\mathrm{C}$ is given by $y^2=x^6-x$. 


If $g=3$ and $\mathrm{C}$ is not hyperelliptic (resp. hyperelliptic), then $[\mathrm{C}]$ is a singular point of $\mathrm{M}\_3$ if and only if $\mathrm{Aut(C)}$ is nontrivial (resp. $\mathrm{Aut(C)}$ is not $\mathbf{Z}/2\mathbf{Z}$).


If $g\geqslant 4$, then $[\mathrm{C}]$ is a singular point of $\mathrm{M}\_g$ if and only if $\mathrm{Aut(C)}$ is nontrivial. 


An algebro-geometric reference for this result is [this paper by H. Popp](https://core.ac.uk/download/pdf/82439366.pdf).


