  We establish a Morita theorem to construct triangle equivalences between the
singularity categories of (commutative and non-commutative) Gorenstein rings
and the cluster categories of finite dimensional algebras over fields, and more
strongly, quasi-equivalences between their canonical dg enhancements. More
precisely, we prove that such an equivalence exists as soon as we find a
quasi-equivalence between the graded dg singularity category of a Gorenstein
ring and the derived category of a finite dimensional algebra which can be done
by finding a single tilting object.
  Our result is based on two key theorems on dg enhancements of cluster
categories and of singularity categories, which are of independent interest.
First we give a Morita-type theorem which realizes certain $\mathbb{Z}$-graded
dg categories as dg orbit categories. Secondly, we show that the canonical dg
enhancements of the singularity categories of symmetric orders have the
bimodule Calabi-Yau property, which lifts the classical Auslander-Reiten
duality on singularity categories.
  We apply our results to such classes of rings as Gorenstein rings of
dimension at most $1$, quotient singularities, and Geigle-Lenzing complete
intersections, including finite or infinite Grassmannian cluster categories, to
realize their singularity categories as cluster categories of finite
dimensional algebras.
