There do seem to be people thinking about it. I heard Sylvain Cappell lecture a few weeks ago; he spoke about using intersection (co)homology as part of a generalization of characteristic classes to singular varieties.

Here's a recent paper by Cappell-Maxim-Shaneson that he referred to: ["Intersection cohomology invariants of complex algebraic varieties"][1].


  [1]: https://www.ams.org/mathscinet-getitem?mr=2454344