My advisor told me the following:

Let $\Sigma$ be a singular surface over $\mathbb{C}$ whose singularities are all ordinary quadratic, or more generally Duval singularities. Let $\epsilon: S \rightarrow \Sigma$ be the desingularization. Then, writing $\omega_\Sigma$ for the dualizing sheaf of $\Sigma$ and $K_S$ for the canonical sheaf on $S$ we have $$ \epsilon^*\omega_\Sigma = K_S. $$ A reference would be Duval's original paper, but this is quite old. Does anyone know a better one?

We tried Reid's chapters on algebraic surfaces and some other obvious ones but did not find it there.