From the summary: We investigate analytically the linear inviscid instability of an infinitely thin vortex sheet, periodically corrugated with finite amplitude along the spanwise direction. 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 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 for both plane Couette and Poiseuille flows.