Let $X$ be a singular variety. Define the singular category (as in Orlov's paper) as the Verdier quotient of the derived category of coherent sheaves on $X$ modulo the full subcategory of perfect complexes.

For example, there is a quiver description in the case of ADE surface singularities: http://arxiv.org/abs/math/0511155

Are there any other cases do we have results for the singular category? In particular, for higher codimension varieties?

Let $X$ be a singular variety. Define the (triangulated) category of singularities (as in Orlov's paper) as the Verdier quotient of the derived category of coherent sheaves on $X$ modulo the full subcategory of perfect complexes.

For example, there is a quiver description in the case of ADE surface singularities: http://arxiv.org/abs/math/0511155

Are there any other cases do we have results for the category of singularities? In particular, for higher codimension varieties?