In this paper, a general method of regular perturbation for linear eigenvalue problems is presented, in which the orders of perturbation terms are extended to infinity. The method of regular perturbation is applied to study vibration mode localization in randomly disordered weakly coupled two-dimensional cantilever-spring arrays. Localization factors, which characterize the average exponential rates of decay or growth of the amplitudes of vibration, are defined in terms of the angles of orientation. First-order approximate results of the localization factors are obtained using a combined analytical-numerical approach. For the systems under consideration, the direction in which vibration is originated corresponds to the smallest localization factor; whereas the “diagonal” directions correspond to the largest rate of decay or growth of the amplitudes of vibration. When plotted in the logarithmic scale, the vibration modes are of a hill shape with the amplitudes of vibration decaying linearly away from the cantilever at which vibration is originated.