The use of an asymptotically valid interface equation for directional solidification allows numerical studies of the evolution of three-dimensional cellular structures in extended systems. We consider systems that are large enough to render a statistical description of disordered structures meaningful and to enable a direct comparison with experiment. Moreover, it is possible to assess the stability of the observed patterns on the basis of long-time simulations. In addition to statistical methods already employed in the analysis of experiments, new statistical tools are introduced to follow the dynamics of the system. In general, three growth phases can be distinguished. During the first, short one, the pattern dynamically selects its preferred length scale by a coarsening or a fine-graining process. In the second, much longer phase, the cells rearrange, evolving towards a polycrystalline, essentially ordered structure. In the third phase, a process of gradual elimination of defects takes place. For smaller temperature gradients, there is an evolution towards oscillating patterns. Oscillations lead to a reduction of the percentage of defects, unless they act as a precursor to weak turbulence, which happens at even lower values of the temperature gradient.