The linear stability of the Stokes layer generated by an oscillating flat plate is investigated using Floquet theory. The results obtained include the behaviour of the growth rate of the disturbances, part of the corresponding neutral curve and the structure of neutrally stable disturbances. Previously unknown properties of the growth rate cause the neutral curve to have a complicated geometry: the majority of the marginal curve is defined by waves propagating relative to the basic flow and the curve is smooth in character, but for certain very narrow bands of wavenumbers it was found that stationary modes are the first to become unstable. This phenomenon has the consequence that the underlying smooth neutral curve is punctuated by thin finger-like features. The structure of the eigenfunctions showed that the neutrally stable disturbances tend to grow most rapidly just after the wall velocity passes through zero.