On the basis of a detailed stochastic simulation we find that nonequilibrium growth in 2+1 dimensions, within the simple solid-on-solid approximation, is surprisingly rich with its dynamical universality depending sensitively on the local atomistic relaxation rules of the growth model. We establish connections between our computed dynamical growth exponents for various physically plausible local growth models with those given by a recently proposed fourth-order nonlinear continuum diflferential equation.PACS numbers: 61.50. Cj, 05.40.+J, 05.70.Ln, 68.55.Bd In a solid-on-solid (SOS) growth model, incident atoms stick only to the tops of the already existing substrate atoms which are arranged in a lattice [1]. The resultant growing structure is, therefore, a lattice of columns whose heights increase as matter is added from outside. The SOS model has been extensively used in both equilibrium and nonequilibrium crystal growth studies. While being conceptually simple, it also describes well many real situations where vacancies and overhangs are relatively rare. In this Letter, we present detailed simulation results which demonstrate that nonequilibrium growth, even within the simple SOS approximation, is extremely rich in 2+1 dimensions (i.e., two-dimensional substrate, ^'=2, with growth in the other direction), leading to many different growth universality classes depending upon the precise nature of the local growth rules. We find that the local atomistic relaxation rule and, in particular, its dependence on the atomic coordination number which defines the model determine the universality class of the growth model. This surprising richness in the SOS nonequilibrium growth is totally unanticipated, because the simplicity of the model makes one expect some simple universality independent of the details of the local growth rules. In fact, it is interesting to note that more complicated nonequilibrium growth models involving ballistic deposition (where impinging atoms may stick to the sides of the existing columns, thus allowing overhangs and vacancies) are all believed to belong to a single universality class, namely, the KPZ universality [2], whereas the simpler SOS growth models cannot be characterized by a single universality class.We impose two additional restrictions on the SOS model, motivated both by practical relevancy and computational tractability, namely, neglect of desorption and, in some cases [3] as discussed below, of any upward atomic relaxation [4]. The continuum equation governing the dynamics of growth under these conservative (i.e., no desorption and SOS restrictions) conditions has the general form [5] bh/bt^dSl^h-bS^h + (fourth-order nonlinear terms) +where rjixj) is the Gaussian white noise satisfying the correlation {T](xj)7]{xj'))^D8 (x-x)d(t-t') with x and x' the lateral coordinates. Since there is no evaporation [6], vacancies [7], or a tilted substrate [8], the (Vh)^ term of the KPZ equation does not appear, as the growth process is manifestly current conserving. In the presence of the ...