The density of states, which measures the density of the spectrum, is evaluated for a platonic crystal (periodically structured elastic plate) using the Green's function approach. Results are presented not only for the standard density of states, but also for the mutual, local, and spectral density of states. These other state functions provide a pathway to the standard density of states and characterize the radiative and other properties of the crystal. This is the first known examination of the density of states for a platonic crystal and extends the existing Green's function approach for photonic crystals to thin, elastic plates. As a motivating example the theory is applied to the problem of a square array of pins embedded in a thin plate. The density of states functions for an empty lattice (a uniform plate) are also presented in order to give a clear illustration of the steps in the derivation. Careful numerical calculations are given which reveal the complex behavior of the crystal, including intervals of suppressed density of states. These results are compared to calculations for a finite crystal with an interior source, and the behaviors of the finite and infinite systems are shown to be connected through the density of states.