Nanometric metal structures exhibit unique optical properties that originate from coherent oscillations of the free electron plasma of metals which are called localized surface plasmon modes. Excitation of the latter through interaction with light comes with a resonant response in the optical absorption and scattering. The spectral profile of such plasmon resonances is thereby strongly dependent on the structure's size and shape as well as on the refractive index of the surrounding medium.The present work describes the plasmonic properties of gold nanostructures featuring a closed ring or split ring geometry. Electron beam lithography was employed to tailor these structures on the nanometer scale. Experimental studies based on optical extinction and scattering spectroscopy were corroborated by numerical simulations using a boundary element method. The discussed structures support multiple localized surface plasmon modes, which is evident from pronounced resonances in the optical extinction and scattering spectra of a single structure as well as of structure ensembles. We found that the modes for both discussed geometries follow the same dispersion relation and that they can be uniformly interpreted in terms of a standing (plasmon) wave model. However, for the ring structure, an excitation of the energetically higher modes (with respect to the dipolar, fundamental mode) is achieved only for an illumination at oblique incidence. Furthermore, the excitation of certain split ring modes is accompanied with remarkable field enhancement values in the gap region. Finally, in an experimental proof of principle, we demonstrate that the split ring structures allow to recognize DNA-oligonucleotide binding events with high sensitivity, as compared to ring and disk shaped structures.