The purpose of my talk will be to explore the geometry and topology of Hilbert schemes of points of surfaces with ADE singularities, guided by the long-established results for the case of smooth surfaces. I will in particular report on a partly conjectural formula for their Euler characteristics, obtained in joint work with Ádám Gyenge and András Némethi. The basic representation of the affine Lie algebra corresponding to the surface singularity via the McKay correspondence, and its crystal basis theory, play an important role in our approach. The resulting function is modular, confirming a new case of S-duality.