We consider the motion of a single quasi-geostrophic ellipsoid of uniform potential vorticity in equilibrium with a linear background shear flow. This motion depends on four parameters: the height-to-width aspect ratio of the vortex, $h/r$, and three parameters characterizing the background shear flow, namely the strain rate, $\gamma$, the ratio of the background rotation rate to the strain, $\beta$, and the angle from which the shear is applied, $\theta$. We generate the equilibria over a large range of these parameters and analyse their linear stability. For the second-order ($m\,{=}\,2$) modes which preserve the ellipsoidal form, we are able to derive equations for the eigenmodes and growth rates. For the higher-order modes we use a numerical method to determine the full linear stability to general disturbances ($m\,{>}\,2$).

Overall we find that the equilibria are stable over most of the parameter space considered, and where instability does occur the marginal instability is usually ellipsoidal. From these results, we determine the parameter values for which the vortex is most stable, and conjecture that these are the vortex characteristics which would be the most commonly observed in turbulent flows.