We investigate Ising ferrimagnets on square and simple cubic lattices with exchange couplings between spins of values S = 1/2 and 1 on neighbouring sites and an additional single-site anisotropy term on the S = 1 sites. Mainly on the basis of a careful and comprehensive Monte Carlo study, we conclude that there is no tricritical point in the two-dimensional case, in contradiction to mean-field predictions and recent series results. However, evidence for a tricritical point is found in the three-dimensional case. In addition, a line of compensation points is found for the simple cubic, but not for the square lattice.