Separation of a supersonic boundary layer near a compression ramp is considered in the limit of large Reynolds numbers and for Mach numbers $O(1)$. When the ramp angle is small, the motion may be described by the well-known triple-deck theory describing viscous–inviscid interactions. For small values of the scaled ramp angle, steady stable solutions can be obtained. However, it is shown that when a recirculation zone is present and the ramp angle is sufficiently large, the flow in the recirculation zone is susceptible to convective instabilities when perturbations are introduced there. At still larger values of the scaled ramp angle, an absolute instability is shown to occur that leads to a violent local breakdown of the boundary layer. The calculated results are shown to be consistent with a theoretical criterion that is the necessary and sufficient condition for the onset of instability.