The linear inviscid instability of an infinitely thin vortex sheet, periodically corrugated with finite amplitude along the spanwise direction, is investigated analytically. Two types of corrugations are studied, one of which includes the presence of an impermeable wall. Exact eigensolutions are found in the limits of very long and of very short wavelengths. The intermediate-wavenumber range is explored by means of a second-order asymptotic series and by limited numerical integration. The sheets are unstable to both sinuous and varicose disturbances. The former are generally found to be more unstable, although the difference only appears for finite wavelengths. The effect of the corrugation is shown to be stabilizing, although in the wall-bounded sheet the effect is partly compensated by the increase in the distance from the wall. The controlling parameter in that case appears to be the minimum separation from the sheet valley to the wall. The instability is traced to a pair of oblique Kelvin–Helmholtz waves in the flat-sheet limit, but the eigenfunctions change character both as the corrugation is made sharper and as the wall is approached, becoming localized near the crests and valleys of the corrugation. The study is motivated by the desire to understand the behaviour of lifted low-speed streaks in wall-bounded flows, and it is shown that the spatial structure of the fundamental sinuous eigenmode is remarkably similar to previously known three-dimensional nonlinear equilibrium solutions in both plane Couette and Poiseuille flows.