The upper-branch neutral modes of inviscid instability in a boundary-layer flow with significant longitudinal vortices present are shown to possess typically a logarithmically singular, non-inflectional, critical layer. This contrasts with previous linear and nonlinear suggestions implemented in vortex–wave interaction and secondary instability theories, which are re-examined. The analysis here is based first on perturbation techniques applied to a Rayleigh unstable planar motion supplemented by a vortex centred around the inflection level, followed by the extension to more general cases. Flows with order one and larger spanwise scales are considered. Multiple solutions, their limit properties and parametric continuations are illustrated with concrete examples.