  We construct ALE Calabi-Yau metrics with cone singularities along the
exceptional set of resolutions of $\mathbb{C}^n / \Gamma$ with non-positive
discrepancies. In particular, this includes the case of the minimal resolution
of two dimensional quotient singularities for any finite subgroup $\Gamma
\subset U(2)$ acting freely on the three-sphere, hence generalizing
Kronheimer's construction of smooth ALE gravitational instantons. Finally, we
show how our results extend to the more general asymptotically conical setting.
