The relative performance of four variants of the Moller-Plesset (MP) partitioning (using different diagonal one-electron unperturbed Hamiltonian, H-0) based state-specific multireference perturbation theory (SS-MRPT) [termed as SS-MRPT(MP)] has been investigated and demonstrated by calculations of the dissociation potential energy curves (PECs) of the first three electronic states [ground state X-3 Sigma(-)(g) as well as low-lying singlet excited states, a(1)Delta(g) and b(1)Sigma(+)(g)] of the oxygen molecule using different basis sets. The spectroscopic constants extracted from the computed PECs obtained by the SS-MRPT(MP) method are calibrated with respect to the corresponding value of the full configuration interaction (FCI) and experimental data for the corresponding states. We have also computed vertical excitation (or transition) energies and compared those with the corresponding FCI values along with the results of other available sophisticated methods. Encouraging agreement between SS-MRPT(MP) theory and some benchmark calculations has been observed. We have thus assessed the applicability and accuracy of the SS-MRPT(MP) method with different diagonal one-electron partitioning schemes. The ability of the SS-MRPT(MP) method with different partitioning schemes to predict full PECs and spectroscopic constants of the ground state and excited states with almost equivalent accuracy is promising.