The linear stability of a laterally sheared barotropic flow along a trench is analyzed. The depth and mean flow profiles used in the model are piecewise constant and piecewise linear, respectively. As a consequence the stream function and dispersion relation describing the perturbation field can be written in terms of elementary functions. The dispersion relation, being a quartic in the wave frequency, yields four neutral wave modes: two shear waves, a shelf wave and a trench wave. An instability arises when two shear wave dispersion curves coalesce or when one of the shear wave curves coalesces with either a shelf or a trench wave dispersion curve. Further, because of the quartic nature of the dispersion relation, the stability diagrams can have overlapping regions of instability in Rossby number—wavenumber space.