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We establish a Morita theorem to construct triangle equivalences between thesingularity categories of (commutative and non-commutative) Gorenstein ringsand the cluster categories of finite dimensional algebras over fields, and morestrongly, quasi-equivalences between their canonical dg enhancements. Moreprecisely, we prove that such an equivalence exists as soon as we find aquasi-equivalence between the graded dg singularity category of a Gorensteinring and the derived category of a finite dimensional algebra which can be doneby finding a single tilting object.  Our result is based on two key theorems on dg enhancements of clustercategories and of singularity categories, which are of independent interest.First we give a Morita-type theorem which realizes certain $\mathbb{Z}$-gradeddg categories as dg orbit categories. Secondly, we show that the canonical dgenhancements of the singularity categories of symmetric orders have thebimodule Calabi-Yau property, which lifts the classical Auslander-Reitenduality on singularity categories.  We apply our results to such classes of rings as Gorenstein rings ofdimension at most $1$, quotient singularities, and Geigle-Lenzing completeintersections, including finite or infinite Grassmannian cluster categories, torealize their singularity categories as cluster categories of finitedimensional algebras.

Enhanced Auslander-Reiten duality and Morita theorem for singularity categories