  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.
