Responses of tectonic plates of the Earth's crust to spatially localized perturbations are a classical problem of theoretical seismology. A characterization of such responses is of fundamental importance for the estimation and interpretation of seismograms, for the prediction and analysis of earthquakes. In this paper, we model a plate of the Earth's crust as a vertically stratified elastic waveguide of finite thickness and infinite horizontal extension rigidly attached to a solid half-space underneath it, and study the propagation of spatially localized linear perturbations in the model. An initial boundary–value linear stability problem for three-dimensional (3D) small localized disturbances in the plate is treated by using the Laplace transform in time and the Fourier transform in two orthogonal spatial directions. By applying an energytype method it is shown that a plate of the Earth's crust of an arbitrary vertical stratification is exponentially neutrally stable. The asymptotic time responses of the plate to sources nearly harmonic in time are studied by applying the mathematical formalism for 3D spatially amplifying waves to the solution having the form of an inverse Laplace-Fourier integral. The procedure is an extension of previous analysis by Brevdo to the vertically stratified case. The vertical stratification of the plate is modelled by using different seismic measurement data. We show that every vertically stratified plate of the Earth's crust considered possesses a rich set of resonant frequencies which is assessed to be countable and unbounded. Sources with resonant frequencies cause a resonant destabilization of the plate, with the growth of the perturbation displacement in time like ln t or √t. The result gives further support to our hypothesis that certain earthquakes can be triggered by localized low-amplitude oscillatory forcings at resonant frequencies. Also, based on the presence of very high resonant frequencies in every tectonic plate, we suggest that the estimation of seismograms and earthquake prediction can be improved by extending the frequency range of the existing monitoring tools.