We present a joint experimental and theoretical investigation of the magnetization reversal process in an array of circular permalloy nanorings. The longitudinal hysteresis loop measured by magneto-optic Kerr effect exhibits the typical two-step switching behavior corresponding to transition from the onion to the vortex state and vice versa with plateaux in each loop branch. The frequencies of the spin excitations in these configurations calculated by means of the dynamical matrix approach are in good agreement with those measured by Brillouin light scattering. Several types of spin modes are found localized in different regions of the rings. It is found that the onion-to-vortex and the vortex-to-onion transitions are triggered by soft spin modes. The shape and symmetry of the dynamic magnetization profile of these soft modes provide the initial modifications of the magnetization at the onset of the transitions.