  We study singularity categories of exact categories with a focus on those
associated to a complete hereditary cotorsion pair. As an application we
identify a non-affine analogue of the singularity category of a Gorenstein
local ring; with this Buchweitz's classic equivalence of three categories over
Gorenstein local rings has been generalized to schemes, a project started by
Murfet and Salarian more than ten years ago. As another application we use the
framework to characterize rings of finite finitistic dimension.
