The field-orientation dependence of magnetization reversal and spin wave modes in nanorings has been investigated by focused magneto-optical Kerr effect spectroscopy, Brillouin spectroscopy, and micromagnetic simulations. The samples studied are periodic arrays of 150 nm wide Ni80Fe20 rings with long axes of 850, 1200, and 1500 nm, and respective short axes of 550, 700, and 900 nm. For the smallest ring, the onion-to-vortex transition is observed for both the applied magnetic field directed along, and 5° from its magnetization easy axis. In contrast for the two larger rings, this transition is not observed for the field in the latter orientation. Splitting of spin wave modes is observed for the transition from the onion to vortex state. Micromagnetic simulations of magnetization reversal and spin waves are in good agreement with experiments. The simulated temporal recording of magnetization distributions during transition reveals that different switching mechanisms result from different applied magnetic field orientations.