We study analogues of the Rayleigh waves in a semiinfinite Kirchhoff plate with periodic edges (the Neumann problem for the biharmonic operator) and in an infinite plate with a periodic family of foreign inclusions. In the first case, we prove the existence of localized waves (exponentially decaying in the perpendicular direction) for any edge profile. In the second case, we obtain a sufficient condition for trapping of waves. We explain why the results obtained for the biharmonic operator are different from the known results for the Helmholtz equation. In the case of threshold resonance, we construct asymptotic expansions of eigenvalues of the model problem, which generate analogues of the Stoneley waves. For a plate with a periodic family of cracks perpendicular to the straight boundary of the half-plane, we detect periodic localized Floquet waves that do not transfer energy.