Hard-sphere model systems are well-suited in both experiment and simulations to investigate fundamental aspects of the crystallization of fluids. In experiments on colloidal models of hard-sphere fluids, the fluid is unavoidably in contact with the walls of the sample cell, where heterogeneous crystallization may take place. In this work we use simulations to investigate the competition between homogeneous and heterogeneous crystallization. We report simulations of wall-induced nucleation for different confining walls. Combining the results of these simulations with earlier studies of homogeneous crystallization allows us to assess the competition between homogeneous and heterogeneous nucleation as a function of the wall type, fluid density and the system size. On flat walls, heterogeneous nucleation will typically overwhelm homogeneous nucleation. Even for surfaces randomly coated with spheres with a diameter that was some three times larger than that of the fluid spheres - as has been used in some experiments - heterogeneous nucleation is likely to be dominant for volume fractions smaller than ∼0.535. Only for a disordered coating that has the same structure as the liquid did we find that nucleation was likely to occur in the bulk. Hence, such coatings might be used to suppress heterogeneous nucleation in experiments. Finally, we report the apparent homogeneous nucleation rate taking into account the formation of crystallites both in the bulk and at the walls. We find that the apparent overall nucleation rates coincide with those reported in "homogeneous nucleation" experiments. This suggests that heterogeneous nucleation at the walls could partly explain the large discrepancies found between experimental measurements and simulation estimates of the homogeneous nucleation rate.