  We study the relationship between singularity categories and relative
singularity categories and discuss constructions of differential graded
algebras of relative singularity categories. As consequences, we obtain
structural results, which are known or generalise known results, on singularity
categories of algebras with radical square zero, of non-commutative
deformations of Kleinian singularities, of $SL_3(\mathbb{C})$-quotient
singularities and of Gorenstein toric threefolds.
