I describe the low-energy worldvolume theory for transverse D-branes in a type IIB compactification near and orbifold singularity, incorporating the possibility of discrete torsion. I study the moduli space of vacuum configurations for various (projective) representations of the quotient group and their relation to the background geometry. I also discuss the deformations of the singularity which arise in string theory and their realization in the worldvolume theory. Extending to this case the known relation between D-brane charges and K-theory I propose a definition of a "stringy" K-theory for these spaces and use it to study certain aspects of their homology. These considerations raise an interesting puzzle concerning the "S-duality" of type-IIB string theory. The talk is based on joint work with P. Aspinwall hep-th/0009042.