Three-dimensional quantum and classical dynamical calculations of the dissociative adsorption of hydrogen have been performed, in which, besides one reaction path coordinate, the lateral degrees of freedom of the hydrogen center of mass were taken into account. These calculations were compared to results obtained by classical and quantum sudden approximations, which assessed the importance of tunneling, zero-point effects, and also steering in the dissociative adsorption of hydrogen. For energies below the minimum barrier height, tunneling is of course the relevant mechanism for dissociation, but above the minimum barrier height quantization and zero-point effects become more prominent. Zero-point effects suppress the dissociation probability; however, for energies slightly above the minimum barrier height, steering of the particles is only operative in the quantum dynamics and can thereby almost compensate the suppression of the quantum sticking probabilities due to zero-point effects, compared to the classical calculations. The consequences of these findings with respect to the concept of zero-point corrections in order to obtain effective quantum barrier heights are discussed. The results presented in this study should be relevant for the reaction and propagation dynamics in all systems containing hydrogen.