This Letter discusses waves in a rotating magnetized fluid layer, governed by ``shallow-water'' magnetohydrodynamics. Such waves likely exist in the solar tachocline, and we focus on this application. A dispersion relation is derived, giving two branches of waves: Alfvén and magnetogravity. In general, finite Alfvén and magnetogravity waves can propagate without change of shape. However, if the Coriolis force is absent, as on the equator of the tachocline, finite magnetogravity waves steepen and develop singularities in a time τs. It is shown that τs increases monotonically with the ambient magnetic field strength.