We investigate Ising ferrimagnets on square and simple-cubic lattices with exchange couplings between spins of values S=1/2 and S=1 on neighbouring sites and an additional single-site anisotropy term on the S=1 sites. Based mainly on 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.
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