We consider a compressible, inviscid, stratified parallel shear flow, bounded below by a rigid wall and above by a half-space of constant wind speed and temperature. Linear stability analysis shows that this flow is unstable to a family of modes, one of which is the well-known Kelvin-Helmholtz disturbance. The remaining modes, here called resonant modes, undergo little attention in the region below the shear layer, and their parameters and structure are strongly influenced by the presence of the ground. The stability curves for both types of modes are investigated as functions of the parameters of the back-ground state. For most combinations of parameters, the resonant modes are trapped between their critical level and the ground. However, for nearly isentropic shear layers, the neutral resonant modes become free to propagate in the upper half-space. Under such conditions, the growing solutions are no longer contiguous to the neutral curves. The growth rates of the Kelvin-Helmholtz modes are found to be larger than those of the resonant modes for all combinations of the background parameters. The evolution of instabilities in a real shear layer is discussed in the light of this result. The eigenfunction structure of the ,resonant modes suggests an explanation for the multiple thin scattering layers often recorded by radars observing the stable boundary layer.