  Let $\X$ be a resolving subcategory of an abelian category. In this paper we
investigate the singularity category
$\ds(\underline\X)=\db(\mod\underline\X)/\kb(\proj(\mod\underline\X))$ of the
stable category $\underline\X$ of $\X$. We consider when the singularity
category is triangle equivalent to the stable category of Gorenstein projective
objects, and when the stable categories of two resolving subcategories have
triangle equivalent singularity categories. Applying this to the module
category of a Gorenstein ring, we characterize simple hypersurface
singularities of type $(\a_1)$ as complete intersections over which the stable
categories of resolving subcategories have trivial singularity categories. We
also generalize several results of Yoshino on totally reflexive modules.
