The restriction of deformations to a subregion of a system undergoing either free or forced vibration due to an irregularity or discontinuity in it is called mode localization. Here, we study mode localization in free and forced vibration of monolithic and unidirectional fiber-reinforced rectangular linearly elastic plates with edges either simply supported (SS) or clamped by using a third-order shear and normal deformable plate theory (TSNDT) with points on either one or two normals to the plate midsurface constrained from translating in all three directions. The plates studied are symmetric about their midsurfaces. The in-house developed software based on the finite element method (FEM) is first verified by comparing predictions from it with either the literature results or those computed by using the linear theory of elasticity and the commercial FE software abaqus. New results include: (i) the localization of both in-plane and out-of-plane modes of vibration, (ii) increase in the mode localization intensity with an increase in the length/width ratio of a rectangular plate, (iii) change in the mode localization characteristics with the fiber orientation angle in unidirectional fiber reinforced laminae, (iv) mode localization due to points on two normals constrained, and (iv) the exchange of energy during forced harmonic vibrations between two regions separated by the line of nearly stationary points that results in a beats-like phenomenon in a subregion of the plate. Constraining translational motion of internal points can be used to design a structure with vibrations limited to its small subregion and harvesting energy of vibrations of the subregion.