Following the investigation of the long-time limit of the impulse response of an incompressible swept boundary layer (Taylor & Peake 1998), we now consider the corresponding behaviour of two representative sets of compressible swept-wing profiles, one set in subsonic flow and the other in supersonic flow. The key feature of the incompressible analysis was the occurrence of modal pinch points in the cross-flow wavenumber plane, and in this paper the existence of such pinches over a wide portion of space in high-speed flow is confirmed. We also show that close to the attachment line, no unstable pinches in the chordwise wavenumber plane can be found for these realistic wing profiles, contrary to predictions made previously for incompressible flow with simple Falker–Skan–Cooke profiles (Lingwood 1997). A method for searching for absolute instabilities is described and applied to the compressible boundary layers, and we are able to confirm that these profiles are not absolutely unstable. The pinch point property of the compressible boundary layers is used here to predict the maximum local growth rate achieved by waves in a wavepacket in any given direction. By determining the direction of maximum amplification, we are able to derive upper bounds on the amplification rate of the wavepacket over the wing, and initial comparison with experimental data shows that the resulting N-factors are more consistent than might be expected from existing conventional methods.