This paper presents solutions to a number of problems posed for the out-of-plane displacement of infinite thin elastic plates that are rigidly pinned in periodic configurations, but that possess a finite number of `defects'. We begin by considering a single one-dimensional periodic array of pins. We derive an analytic solution for the displacement produced by the forced oscillation of the central pin in the array, and this solution is shown to be closely connected to the problem of scattering of plane waves by an array when a finite number of pins are removed. Attention then focuses on doubly-periodic rectangular arrays of pinned points possessing defects. Central to approaching such problems is an understanding of Bloch-Floquet waves in periodic arrays in the absence of defects and a simple method is described for computing the associated dispersion surfaces. The solution to three problems are then sought: the trapping of localized waves by a finite number of missing pins; trapping of waves by entire rows of missing pins; and the wave radiation pattern due to the forcing of a single pin. All problems are treated analytically by using bounded Green's functions for thin elastic plates, a discrete Fourier transform solution method and simple, explicit and rapidly convergent evaluations of the one- and two-dimensional lattice sums that arise.