The dispersive behavior of waves in softening problems is analysed. Attention is focused on the influence of the numerical scheme on the dispersion characteristics in the process of localization of deformation. Distinction has been made between softening models defined in a standard plasticity framework and in a gradient-dependent plasticity theory. For a modelling with and without the inclusion of strain gradients, accuracy considerations with respect to mass discretization, finite element size, time integration scheme and time step have been carried out.