The prediction of molecular properties such as equilibrium structures or vibrational wavenumbers is a routine task in computational chemistry. If very high accuracy is required, however, the use of computationally demanding ab initio wavefunction methods is mandatory. We present property calculations utilizing Retaining the Excitation Degree - Møller-Plesset (REMP) and Orbital Optimized REMP (OO-REMP) hybrid perturbation theories, showing that with the latter approach, very accurate results are obtained at second order in perturbation theory. Specifically, equilibrium structures and harmonic vibrational wavenumbers and dipole moments of closed and open shell molecules were calculated and compared to the best available experimental results or very accurate calculations. OO-REMP is capable of predicting bond lengths of small closed and open shell molecules with an accuracy of 0.2 and 0.5 pm, respectively, often within the range of experimental uncertainty. Equilibrium harmonic vibrational wavenumbers are predicted with an accuracy better than 20 cm-1. Dipole moments of small closed and open shell molecules are reproduced with a relative error of less than 3%. Across all investigated properties, it turns out that a 20%:80% Møller-Plesset:Retaining the Excitation Degree mixing ratio consistently provides the best results. This is in line with our previous findings, featuring closed and open shell reaction energies.