We use a microscopic theory taking into account dipolar and nearest-neighbour exchange interactions to explore spin-wave excitations in two-dimensional square-shaped magnetic nanorings with the Landau state assumed as a magnetic state. From the spin-wave spectra, we determine the range of the dipolar-to-exchange interaction ratio in which the assumed state is stable. Various types of localized spin waves prove responsible for the transition to a new magnetic configuration. We found the transition forced by predominating exchange interactions size-independent in a wide range of both external and internal size of the ring. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4745875]