  We obtain, via the formalism of tensor actions, a complete classification of
the localizing subcategories of the stable derived category of any affine
scheme with hypersurface singularities and of any local complete intersection
over a field; in particular this classifies the thick subcategories of the
singularity categories of such rings. The analogous result is also proved for
certain locally complete intersection schemes. It is also shown that from each
of these classifications one can deduce the (relative) telescope conjecture.
