  Let $R$ be an isolated Gorenstein singularity with a non-commutative
resolution $A=End_R(R\oplus M)$. In this paper, we show that the relative
singularity category $\Delta_R(A)$ of $A$ has a number of pleasant properties,
such as being Hom-finite. Moreover, it determines the classical singularity
category $D_{sg}(R)$ of Buchweitz and Orlov as a certain canonical quotient
category. If $R$ has finite CM type, which includes for example Kleinian
singularities, then we show the much more surprising result that $D_{sg}(R)$
determines $\Delta_R(Aus(R))$, where $Aus(R)$ is the corresponding Auslander
algebra. The proofs of these results use dg algebras, $A_\infty$ Koszul
duality, and the new concept of dg Auslander algebras, which may be of
independent interest.
