In this contribution, the vapor-liquid equilibria (VLE) of binary mixtures containing Lennard-Jones (LJ), Stockmayer (LJ + point dipole, ST), and shifted Stockmayer (sST) fluids are investigated with molecular simulations. The LJ and ST fluids are commonly used nonpolar and polar model fluids, respectively. The sST fluid differs from the ST fluid in that the dipole is shifted away from the dispersive center, which results in the sST fluid exhibiting H-bonding-like behavior. By varying the dispersion energy of the LJ fluid and the dipole moments of the ST and sST fluids, and the dipole shift of the sST fluids, different mixtures are generated, which are investigated at several temperatures and concentrations. Differences in the phase behavior between mixtures containing ST and sST fluids are quantified and linked to differences in fluid structure. The reported data include the VLE envelopes, coexisting densities, activity coefficients, and enthalpies for liquid and vapor phases, as well as structural information on the fluids’ intermolecular orientations.