We have observed localized spin-wave modes in individual thin-film ferromagnetic wires using time-resolved Kerr microscopy as a micron-scale spectroscopic probe. The localization is due to the partial demagnetization of a wire when an external field is applied in the plane of the film and perpendicular to the long axis of the wire. Spatially-resolved spectra demonstrate the existence of distinct modes at the edges of a rectangular wire. Spectral images clearly show the crossover of the two edge modes into a single mode in low applied fields, in agreement with the results of micromagnetic simulations.