  Let $R$ be a right notherian ring. We introduce the concept of relative
singularity category $\Delta_{\mathcal{X}}(R)$ of $R$ with respect to a
contravariantly finite subcategory $\mathcal{X}$ of $\rm{mod}\mbox{-}R.$ Along
with some finiteness conditions on $\mathcal{X}$, we prove that
$\Delta_{\mathcal{X}}(R)$ is triangle equivalent to a subcategory of the
homotopy category $\mathbb{K}_{\rm{ac}}(\mathcal{X})$ of exact complexes over
$\mathcal{X}$. As an application, a new description of the classical
singularity category $\mathbb{D}_{\rm{sg}}(R)$ is given. The relative
singularity categories are applied to lift a stable equivalence between two
suitable subcategories of the module categories of two given right notherian
ring to get a singular equivalence between the rings. In different types of
rings, including path rings, triangular matrix rings, trivial extension rings
and tensor rings, we provide some consequences for their singularity
categories.
