We have expanded our earlier Monte Carlo model [Phys. Rev. A 38, 2447 (1988); J. Crystal Growth 100, 313 (1990)] to three dimensions and included reevaporation after accommodation and growth on dislocation-induced steps. We found again that, for a given set of growth parameters, the critical size, beyond which a crystal cannot retain its macroscopically faceted shape, scales linearly with the mean free path in the vapor. However, the three-dimensional (3D) the systems show increased shape stability compared to corresponding 2D cases. Extrapolation of the model results to mean-free-path conditions used in morphological stability experiments leads to order-of-magnitude agreement of the predicted critical size with experimental findings. The stability region for macroscopically smooth (faceted) surfaces in the parameter space of temperature and supersaturation depends on both the surface and bulk diffusion. While surface diffusion is seen to smooth the growth morphology on the scale of the surface diffusion length, bulk diffusion is always destabilizing. The atomic surface roughness increases with increase in growth temperature and supersaturation. That is, the tendency of surface kinetics anisotropies to stabilize the growth shape is reduced through thermal and kinetic roughening. It is also found that the solid-on-solid assumption, which can be advantageously used at low temperatures and supersaturations, is insufficient to describe the growth dynamics of atomically rough interfaces where bulk diffusion governs the process. For surfaces with an emerging screw dislocation, we find that the spiral growth mechanism dominates at low temperatures and supersaturations. The polygonization of a growth spiral decreases with increasing temperature or supersaturation. When the mean free path in the nutrient is comparable to the lattice constant, the combined effect of bulk and surface diffusion reduces the terrace width of a growth spiral in its center region. At elevated temperatures and supersaturations, 2D nucleation-controlled growth can dominate in corner and edge regions of a facet, while the spiral growth mode prevails in its center. Thus, in addition to confirming the experimental observation that the critical size of a growing crystal depends on the prevailing growth mechanism, we are able to obtain detailed insight into the processes leading to the loss of face and facet stability. © 1991 The American Physical Society.