The moduli space of stable $2$-pointed genus $1$ curves has cyclic quotient singularities, so any Weil divisor on $\overline{M}_{1,2}$ is $\mathbb{Q}$-Cartier.
Is one of the two boundary divisors $\Delta_{irr}$ and $\Delta_{0,2}$ of $\overline{M}_{1,2}$ Cartier ?
The singularities of $\overline{M}_{1,2}$ are located as follows:
a singularity of type $\frac{1}{4}(2,3)$ representing an elliptic curve of Weierstrass representation $C_{4}$ with marked points $[0:1:0]$ and $[0:0:1]$;
a singularity of type $\frac{1}{3}(2,4)$ representing an elliptic curve of Weierstrass representation $C_{6}$ with marked points $[0:1:0]$ and $[0:1:1]$.
a singularity of type $\frac{1}{6}(2,4)$ representing a reducible curve whose irreducible components are an elliptic curve of type $C_{6}$ and a smooth rational curve connected by a node;
a singularity of type $\frac{1}{4}(2,6)$ representing a reducible curve whose irreducible components are an elliptic curve of type $C_{4}$ and a smooth rational curve connected by a node.
The divisor $\Delta_{irr}$ is contained in the smooth locus of $\overline{M}_{1,2}$ and it is Cartier, while the other boundary divisor contains two singular points, it is $\mathbb{Q}$-Cartier but not Cartier.
