Motivated by several biological models such as the Tuckwell-Miura model describing cortical spreading depression, we investigate the types of wave solutions that can exist for reaction-diffusion systems of two equations in which the reaction terms are degenerate in the sense that they are linearly dependent. In particular, we show that there are surprising differences between the types of waves that occur in a single reaction-diffusion equation and the types of waves that occur in a degenerate system of two equations. Importantly, and in contrast to previously published results, we demonstrate that non-stationary pulse solutions can exist for a degenerate system of two equations but cannot exist for a single reaction-diffusion equation. We show that this has important consequences for the minimal model that can generate the types of waves observed in cortical spreading depression. We also give an explicit condition on the source term in a degenerate reaction-diffusion system that guarantees the existence of non-stationary and stationary pulse solutions. We use this approach to provide several examples of reaction terms that have analytical pulse solutions. We also show that the case in which one species cannot diffuse is singular in the sense that the degenerate reaction-diffusion system can admit infinite families of stationary piecewise constant solutions. We further show how such solutions can be accessed by perturbing a constant rest state with a localised continuous disturbance.