In this work we prove the generic simplicity of the spectrum of the clamped plate equation in a bounded regular domain of R-d. That is, given Omega subset of R-d, we show that there exists an arbitrarily small deformation of the domain u, such that all the eigenvalues of the plate system in the deformed domain Omega + u are simple. To prove this result we first prove a nonstandard unique continuation property for this system that also holds generically with respect to the perturbations of the domain. Both the proof of this generic uniqueness result and the generic simplicity of the spectrum use Baire's lemma and shape differentiation. Finally, we show an application of this unique continuation property to a result of generic stabilization for a plate system with one dissipative boundary condition.