In the present work, four nontrivial stages of electrokinetic instability are identified by direct numerical simulation (DNS) of the full Nernst-Planck-Poisson-Stokes system: (i) a stage of the influence of the initial conditions (milliseconds); (ii) one-dimensional (1D) self-similar evolution (milliseconds-seconds); (iii) a primary instability of the self-similar solution (seconds); (iv) a nonlinear stage with secondary instabilities. The self-similar character of evolution at moderately large times is confirmed. Rubinstein and Zaltzman instability and noise-driven nonlinear evolution toward overlimiting regimes in ion-exchange membranes are numerically simulated and compared with theoretical and experimental predictions. The primary instability which happens during this stage is found to arrest a self-similar growth of the diffusion layer. It also specifies its characteristic length as was first experimentally predicted by Yossifon and Chang [G. Yossifon and H.-C. Chang, Phys. Rev. Lett. 101, 254501 (2008)]. A novel principle for the characteristic wave-number selection from the broadband initial noise is established.