Differential graded enhancements of singularity categories

The singularity category of a ring detects the homological singularity of the given ring, and appears in many different contexts. We describe two different dg enhancements of the singularity category, that is, the Vogel dg category and the singular Yoneda dg category. These two dg enhancements turn out to be quasi-equivalent. We report some progress on the Singular Presilting Conjecture.