We present a combined experimental, computational, and semianalytical study of the magnetization dynamics of permalloy disks and nanostructured rings with a systematically varied ring width. We investigate the dynamics of the quasisaturated state. In the case of wide rings the spin wave mode spectrum is similar to that of a disk. The small inner hole can be viewed as a weak perturbation. When the central hole increases, its influence becomes more important and two characteristic modes form gradually. They become localized at different positions in the ring. We explain the localization by the increasing inhomogeneity of the internal magnetic field and the formation of domain wall regions as the rings grow more narrow. In narrow rings one of the modes is clearly confined to the two segments where the internal field Hint is at a maximum and the other to the domain wall region where Hint is small. Dynamic magnetic simulations agree well with the measured spectra and confirm this interpretation. We also applied a semianalytical model, which confirms that the mode localization is driven by the internal field inhomogeneity.