references for singularities: does quotient singularities imply gorenstein? Is there a good place where to learn about singularities of algebraic varieties?
OK, this question is terribly vague, so I'll ask what I'm currently interested in: if X is a smooth variety and G is a finite group, then is $X/G$ Gorenstein? If true, what would a reference be?

Here is a related question on mathoverflow.
It sounds like the relevant part of 'Gorenstein' here is that the canonical divisor is Cartier.  Or, in other words, the canonical sheaf is invertible.  This is not necessarily true for quotients of smooth varieties.
Here is a simple example of an isolated quotient singularity which is non-Gorenstein:  Let $X \cong \mathbb{C}^3 = \mathrm{Spec}\,\mathbb{C}[z_1,z_2,z_3]$ and let $G \cong \mathbb{Z}_2$ act on $X$ as $(z_1, z_2, z_3) \to (-z_1, -z_2,-z_3)$.  Then $X/G = \mathrm{Spec}\,\mathbb{C}[z_1^2,z_2^2,z_3^2,z_1 z_2, z_1 z_3, z_2 z_3]$, and this is not Gorenstein.  There are (at least) a couple of ways to see this:


*

*If you know toric geometry, it is a simple exercise to show that $K_{X/G}$ is not Cartier.

*Roughly, the stalk of $\omega_X$ at the origin is generated by $dz_1\wedge dz_2\wedge dz_3$, and this is not preserved by $G$, so $\omega_{X/G\setminus\{0\}}$ cannot be extended to a line bundle at the origin.  In more detail, $\omega_{X/G}$ has three independent sections in a neighbourhood of the origin:
$z_1dz_1\wedge dz_2\wedge dz_3~,~ z_2dz_1\wedge dz_2\wedge dz_3~,~ z_3dz_1\wedge dz_2\wedge dz_3$.

