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The spatial supersonic shear layer in a rectangular channel is investigated by numerical simulation with the piecewise parabolic method, a high-order Godunov-type scheme. Both streams are supersonic with Mach number 4.5 and 1.6, respectively. A weak broadbanded noise is introduced at the inflow boundary and the growth of multiple unstable modes is observed. In order to extract detailed information from the computed data, the time-dependent flow field is Fourier transformed in time and in the spanwise direction. For a given frequency and a given spanwise wave number the complex eigenfunction and streamwise wave number can be compared to results from linear stability theory. This work aims at investigating in detail the limit of the linear development of the flow and to describe the transition to turbulence. In particular, this technique enables one to distinguish between oblique subsonic instabilities of the Kelvin–Helmholtz (KH) type and supersonic wall modes. A good agreement with linear stability theory is obtained in two and three dimensions until a characteristic streamwise distance is reached, where the flow becomes nonlinear. It is shown that it is mainly the oblique KH modes which are responsible for an increase in the spreading rate during transition and that the nonlinear regime in three dimensions is therefore essentially different from that in two dimensions. The most unstable mode in our configuration is a wall mode and the frequency spectra are largely dominated by the wall modes. However, it turns out that the transition is not governed by the wall modes, but by the oblique KH modes, which evolve into longitudinal streaks.