The standard ways of computing, or indeed defining, invariants of singular varietis use the procedure of resolution of singularities. To explicitly know a resolution, however, requires a good understanding of the singularities of the variety in question.
I will talk about methods that can be applied to the study of singular varieties without resolution of singularities, which originate from localization theorems in topology. As an example, I will present some computations involving the torus-equivariant cohomology of the (refined) Hilbert scheme of points on the plane.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/5312