The linear stability of a thin liquid layer bounded from above by a free surface and from below by an oscillating plate is investigated for disturbances of arbitrary wavenumbers, a range of imposed frequencies and selective physical parameters. The imposed motion of the lower wall occurs in its own plane and is unidirectional and time-periodic. Long-wave instabilities occur only over certain bandwidths of the imposed frequency, as determined by a long-wavelength expansion. A fully numerical method based on Floquet theory is used to investigate solutions with arbitrary wavenumbers, and a new free-surface instability is found that has a finite preferred wavelength. This instability occurs continuously once the imposed frequency exceeds a certain threshold. The neutral curves of this new finite-wavelength instability appear significantly more complex than those for long waves. In a certain parameter regime, folds occur in the finite-wavelength stability limit, giving rise to isolated unstable regions. Only synchronous solutions are found, i.e. subharmonic solutions have not been detected. In Appendix A, we provide an argument for the non-existence of subharmonic solutions.