Free oscillations and stability under an axial compression of a thin cylindrical plate with a weakly fixed rectilinear edge made of a transversally isotropic material with low stiffness with respect to transverse displacements are considered. The curvilinear edges of the plate are assumed to be hingedly supported. The oscillation frequencies and the critical load for a plate with a free or weakly fixed edge are smaller than those for a shell closed in the circumferential direction. The shapes of oscillations and the forms of stability loss localized near the weakly fixed edge and damped at a distance from it are considered. The Timoshenko-Reissner model is used. Localized forms are analyzed by using a system of equations for Timoshenko-Reissner shallow shells, which is derived for this purpose. The main special feature of this system is that it contains a separate equation describing a solution with large variability. For the example of the stability problem under consideration, the error involved in the system of equations for Timoshenko-Reissner shallow shells is studied. The critical load values obtained with the use of the Kirchhoff-Love and Timoshenko-Reissner models are compared.