The three-dimensional Rayleigh–Taylor instability is studied using a lattice Boltzmann scheme for multiphase flow in the nearly incompressible limit. This study focuses on the evolution of the three-dimensional structure of the interface. In addition to the bubble and spike fronts, a saddle point is found to be another important landmark on the interface. Two layers of heavy-fluid roll-ups, one at the spike tip and the other at the saddle point, were observed. The secondary instability in the horizontal planes entangles the already complicated structure of the interface. Parallel computations are utilized to accommodate the massive computational requirements of the simulations.