We present a general and effective algebraic framework for enumerating finite-length quotients of a torsion-free sheaf of arbitrary rank (the Quot zeta function) and finite-length coherent sheaves (the Coh zeta function) over reduced singular curves. We prove that Quot zeta functions are motivically rational, using a novel parametrization and the geometry of affine Grassmannians, and that they satisfy an arbitrary-rank reflection principle, via harmonic analysis. We show that the a normalized high-rank limit of Quot zeta functions converges to the Coh zeta function. As a first application, we compute explicit formulas for these zeta functions for all $y^2 = x^n$ singularities, revealing a surprising and previously unknown connection to Rogers--Ramanujan type $q$-series. Further applications to affine Springer fibers and commuting varieties are also discussed.