Let $V$ be a complex vector space with a faithful linear action of a finite group $G$. Viewing $V$ as affine space (with coordinate ring $\mathbb{C}[V]$), the quotient $V/G$ is the affine variety with coordinate ring the invariant ring $\mathbb{C}[V]^G$. Typically, the image of the origin is a singular point of $V/G$. (Not always. If $G$ is a reflection group, the invariant ring is polynomial, so that $V/G$ is isomorphic with affine space.)
If $G$ is abelian, this singular point is said to be abelian. More generally, if $X$ is any complex algebraic variety and $p\in X$ is a singular point, it is an abelian singularity if $\widehat{\mathcal{O}_{X,p}}$ is isomorphic to $\widehat{\mathcal{O}_{V/A,0}}$ for some abelian group $A$ and linear representation $V$ of $A$, where $0\in V/A$ is the image of the origin.
My question is this: suppose $G$ is not abelian. Can it still happen that the singularity at $0\in V/G$ is an abelian singularity in this sense?
If so, what's an example? (Even better, when in general does this happen?) If not, what's the proof?
Addendum: (3/26) I asked a refinement of this question on MO.
The answer to your question can be found in Theorem 1.7 of Geraschenko and Satriano's paper "Torus quotients as global quotients by finite groups"; see
http://arxiv.org/pdf/1201.4807.pdf