  We show that tautological integrals on Hilbert schemes of points can be
written in terms of universal polynomials in Chern numbers. The results hold in
all dimensions, though they strengthen known results even for surfaces by
allowing integrals over arbitrary "geometric" subsets (and their
Chern-Schwartz-MacPherson classes).
  We apply this to enumerative questions, proving a generalised G\"ottsche
Conjecture for all singularity types and in all dimensions. So if L is a
sufficiently ample line bundle on a smooth variety X, in a general subsystem
P^d of |L| of appropriate dimension the number of hypersurfaces with given
singularity types is a polynomial in the Chern numbers of (X,L). When X is a
surface, we get similar results for the locus of curves with fixed "BPS
spectrum" in the sense of stable pairs theory.
