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 $3$ ?**



Singularities of moduli spaces of curves
No. In fact more is true: the locus of all $n$-pointed curves of genus $g-1$ with a single elliptic tail $E$, such that $\mathrm{Aut}(E)=\mathbf Z/6$, has codimension two in $\overline M\_{g,n}$ and consists of *noncanonical* singularities. This was famously determined by Harris and Mumford in their paper on the Kodaira dimension of the moduli space of curves (they work with $n=0$ but this makes no difference).


