  In this article we construct various models for singularity categories of
modules over differential graded rings. The main technique is the connection
between abelian model structures, cotorsion pairs and deconstructible classes,
and our constructions are based on more general results about localization and
transfer of abelian model structures. We indicate how recollements of
triangulated categories can be obtained model categorically, discussing in
detail Krause's recollement for the stable derived category. In the special
case of curved mixed Z-graded complexes, we show that one of our singular
models is Quillen equivalent to Positselski's contraderived model for the
homotopy category of matrix factorizations.
