In a previous paper, we have considered the weakly nonlinear interaction of a pair of axisymmetric lower branch Tollmien–Schlichting instabilities in cylindrical supersonic flows. Here the possibility that non-axisymmetric modes might also exist is investigated. In fact it is found that such modes do exist and, on the basis of linear theory, it appears that these modes are the most important. The non-axisymmetric modes are found to exist for flows around cylinders with non-dimensional radius a less than some critical value ac. This critical value ac is found to increase monotonically with the azimuthal wavenumber n of the disturbance and it is found that unstable modes always occur in pairs. We show that, in general, instability in the form of lower branch Tollmien—Schlichting waves will occur first for non-axisymmetric modes and that in the unstable regime, the largest growth rates correspond to the latter modes.