Properties of singularities that are preserved by categorical quotients

Let $G$ be a reductive group acting on an affine singular variety $X$, and let $X/G$ be the categorical quotient. I know that if $X$ has rational singularities, then so does $X/G$ (http://link.springer.com/article/10.1007%2FBF01405091). I am curious about other properties of singularities that pass to the quotient. For example, if $X$ is Gorenstein, must $X/G$ be Gorenstein? What if $X$ has canonical singularities? More generally, I'd be happy for a reference where these questions are discussed.

Neither Gorenstein nor canonical are preserved. Already Gorenstein is destroyed for the action of $\mathbb{Z}/3\mathbb{Z}$ on $\mathbb{A}^2$ acting by $(x,y)\mapsto (\zeta\cdot x,\zeta\cdot y)$, where $\zeta$ is a primitive cube root of $1$. Also, the whole point of the Reid -- Shepherd-Barron -- Tai criterion is to determine when a quotient singularity is canonical. Typically it is not canonical.