For singular plane curves, there is a conjectural formula (due to Alexei Oblomkov and myself) in terms of the HOMFLY polynomial of the links of the singularities. For curves whose singularities are torus knots, i.e. like x^a = y^b for a,b relatively prime, and for a few more singularities, the conjecture has been established. See this preprint
Vivek Shende.
Edit: More recently I have given a different characterization of these numbers in terms of multiplicities of certain strata in the versal deformation of the singular curve.
