We will start with an introduction to singularity categories, which were first studied by Buchweitz and later rediscovered by Orlov. Then we will explain what is known about triangle equivalences between singularity categories of commutative rings, recalling results of Knörrer, D. Yang (based on our joint works on relative singularity categories. This result also follows from work of Kawamata and was generalized in a joint work with Karmazyn) and a new equivalence obtained in arXiv:2103.06584.

In the remainder of the talk, we will focus on the case of Gorenstein isolated singularities and especially hypersurfaces, where we give a complete description of quasi-equivalence classes of dg enhancements of singularity categories, answering a question of Keller & Shinder. This is based on arXiv:2108.03292.