Let $\overline{M}_{g,n}$ be the moduli space of $n$-pointed genus $g$ Deligne-Mumford stable curves. This is a normal projective scheme. Then $$codim_{\overline{M}_{g,n}}Sing(\overline{M}_{g,n})\geq 2.$$ For instance for $g = 1, n = 2$ wa have that $\overline{M}_{1,2}$ is a rational surface with four singular points.

**Does there exist any value of $g$ and $n$ for which $Sing(\overline{M}_{g,n})$ is in codimension at least $3$ ?**