  Let $G\subseteq GL(n)$ be a finite group without pseudo-reflections. We
present an algorithm to compute and verify a candidate for the Cox ring of a
resolution $X\rightarrow \mathbb{C}^n/G$, which is based just on the geometry
of the singularity $\mathbb{C}^n/G$, without further knowledge of its
resolutions. We explain the use of our implementation of the algorithms in
Singular. As an application, we determine the Cox rings of resolutions
$X\rightarrow \mathbb{C}^3/G$ for all $G\subseteq GL(3)$ with the
aforementioned property and of order $|G|\leq 12$. We also provide examples in
dimension 4.
