Models for singularity categories

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 _Kac(Inj(R))→K(Inj(R))→D(R)${\mathrm{K}}_{\mathrm{ac}}(\mathrm{Inj}(R))\to \mathrm{K}(\mathrm{Inj}(R))\to \mathrm{D}(R)$. In the special case of curved mixed Z$\mathbb{Z}$-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.