In order to resolve some puzzling discrepancies between theoretical and experimental dipole moment data for oxides of the Group IVa metals a series of nonrelativistic and relativistic high-level-correlated calculations have been performed for SnO and PbO, The same methods have been used to calculate the dipole moments of SnS and PbS for which the earlier results were found to be in reasonable agreement with experiment. The main attention of the present study has been focused on the treatment of the electron correlation contribution to dipole moment of highly ionic oxides. This contribution has been found to be strongly method-dependent. A comparison of different single and multiconfiguration methods shows that the coupled cluster approach with iterative treatment of single and double excitations and perturbative correction for the contribution of triple excitations, CCSD(T), offers presumably the most reliable scheme for the evaluation of the electron correlation contribution to dipole moments of the studied molecules. We have also verified the reliability of the earlier approximate (mass-velocity and Darwin terms) treatment of relativistic contributions by using the scalar Douglas-Kroll approximation. The PbO molecule has also been studied at the level of the Dirac-Hartree-Fock approximation with the results revealing a large spin-orbit contribution to its dipole moment. The present most accurate results, calculated by using the scalar Douglas-Kroll scheme and the CCSD(T) approximation for the electron correlation contribution, are found to give the oxide dipole moments systematically too low by about 0.1 a.u. In the case of the PbO molecule they are also compared with recent results obtained in the framework of the pseudopotential formalism.