Many synthetic quantum systems allow particles to have dispersion relations that are neither linear nor quadratic functions. Here, we explore single-particle scattering in general spatial dimension D >= 1 when the density of states diverges at a specific energy. To illustrate the underlying principles in an experimentally relevant setting, we focus on waveguide quantum electrodynamics (QED) problems (i.e., D = 1) with dispersion relation epsilon(k) = +/-vertical bar d vertical bar k(m), where m >= 2 is an integer. For a large class of these problems for any positive integer m, we rigorously prove that when there are no bright zero-energy eigenstates, the S matrix evaluated at an energy E -> 0 converges to a universal limit that is only dependent on m. We also give a generalization of a key index theorem in quantum scattering theory known as Levinson s theorem-which relates the scattering phases to the number of bound states-to waveguide QED scattering for these more general dispersion relations. We then extend these results to general integer dimensions D >= 1, dispersion relations epsilon(k) = vertical bar k vertical bar(a) for a D-dimensional momentum vector k with any real positive a, and separable potential scattering.