The lineshapes of two-dimensional magnetic resonance spectra of disordered or partially ordered solids are dominated by ridges of singularities in the frequency plane. The positions of these ridges are described by a branch of mathematics known as catastrophe theory concerning the mapping of one 2D surface onto another. We systematically consider the characteristics of HYSCORE spectra for paramagnetic centers having electron spin S=1/2 and nuclear spin I=1 in terms of singularities using an exact solution of the nuclear spin Hamiltonian. The lineshape characteristics are considered for several general cases: zero nuclear quadrupole coupling; isotropic hyperfine but arbitrary nuclear quadrupole couplings; coincident principal axes for the nuclear hyperfine and quadrupole tensors; and the general case of arbitrary nuclear quadrupole and hyperfine tensors. The patterns of singularities in the HYSCORE spectra are described for each case.