(Based on joint work with Ruofan Jiang) The Hilbert scheme of points on a variety  parametrizes -dimensional quotients of the structure sheaf. When  is a planar singular curve, its enumerative invariants have close relation to mathematical physics, knot theory and combinatorics. In this talk, we investigate two analogous moduli spaces, one being a direct generalization of the Hilbert scheme. Our results reveal their surprising relations to Hall polynomials, matrix equations, modular forms, etc.