It is known that thin-walled elastic rings and pipes are subject to instability when under a state of compressive stresses. A particularly interesting example is the buckling of an elastic ring under hydrostatic pressure. The buckling load is strongly influenced by the following nature of the force due to the hydrostatic pressure and, if this effect is neglected, the forecast of the critical buckling load can be up to 50 per cent for slender rings. This work studies, using a non-linear variational formulation, the buckling and vibration characteristics of rings and pipes supported by a Pasternak elastic foundation, the Winkler foundation being considered as a particular case. First, the equation of motion of the preloaded ring is derived and the analytical solution of the eigenvalue problems is obtained. Parametric analysis shows the influence of the geometric and physical parameters of the ring and the foundation on the critical load, natural frequencies and nonlinear load-frequency relationship, considering the force following effect of the hydrostatic pressure. Additionally, the effect of the foundation on pre-critical deformations is studied. Finally, the effect of an initial geometric imperfection is assessed using the Galerkin method. The results show that the parameters of the Pasternak foundation have a considerable effect on the critical load and mode as well as on the natural frequencies of the ring.