The asymptotic structure of the discrete spectrum of a compressible inviscid swirling flow with arbitrary radial distributions of density, pressure and velocity is described for disturbances with large wavenumbers. It is shown that discrete eigenmodes are unstable when a criterion derived by Eckhoff & Storesletten (1978) is satisfied. In general, these modes are characterized by a length scale of order $|m|^{-3/4}$ where $|m|\,{\gg}\,1$ is the azimuthal wavenumber of the disturbance. They have a spatial structure similar to the incompressible modes obtained by Leibovich & Stewartson (1983). In the particular case of solid-body rotation with a positive gradient of entropy, the unstable discrete spectrum contains modes which scale with $|m|^{-1/2}$. If the modes are localized near a solid boundary, they scale with $|m|^{-2/3}$.