Let $S$ be a normal algebraic surface. The following are equivalent:

1. (1)

   $S$ has only canonical singularities.

2. (2)

   $S$ is locally analytically isomorphic to $X=\mathbb{C}^{2}/G$, $G\subset SL_{2}(\mathbb{C})$ a finite subgroup.

3. (3)

   The exceptional divisors of the minimal resolution $\pi_{min}$ of $S$, have simple normal crossings, their components are (-2) smooth rational curves, their incidence graphs are of type A-D-E (Du Val singularities).

The log terminal surface singularities are precisely the singularities of the form $X=\mathbb{C}^{2}/G$, $G\subset GL_{2}(\mathbb{C})$ a finite subgroup.