A pattern of localization, called inhomogeneous localization, is found for dipolar eigenmodes (surface plasmons or eigenstates of the corresponding Schrödinger equation) of fractal clusters. At any given frequency, individual eigenmodes are dramatically different from each other, their sizes vary in a wide range, and their internal geometry may be topologically disconnected and singular at the small scale. These properties differ principally from the results reported for vibrational eigenmodes of fractals, which is attributed to the long-range interaction and non-Goldstonian nature of the polar modes.