© 2018 American Chemical Society. We derive and assess two new classes of regularizers that cope with offending denominators in the single-reference second-order Møller-Plesset perturbation theory (MP2). In particular, we discuss the use of two types of orbital energy dependent regularizers, κ and σ, in conjunction with orbital-optimized MP2 (OOMP2). The resulting fifth-order-scaling methods, κ-OOMP2 and σ-OOMP2, have been examined for bond-breaking, thermochemistry, nonbonded interactions, and biradical problems. Both methods with strong enough regularization restore restricted to unrestricted instability (i.e., Coulson-Fischer points) that unregularized OOMP2 lacks when breaking bonds in H2, C2H6, C2H4, and C2H2. The training of the κ and σ regularization parameters was performed with the W4-11 set. We further developed scaled correlation energy variants, κ-S-OOMP2 and σ-S-OOMP2, by training on the TAE140 subset of the W4-11 set. Those new OOMP2 methods were tested on the RSE43 set and the TA13 set where unmodified OOMP2 itself performs very well. The modifications we made were found insignificant in these data sets. Furthermore, we tested the new OOMP2 methods on singlet biradicaloids using Yamaguchi's approximate spin-projection. Unlike the unregularized OOMP2, which fails to converge these systems due to the singularity, we show that regularized OOMP2 methods successfully capture strong biradicaloid characters. While further assessment on larger data sets is desirable, κ-OOMP2 with κ = 1.45 Eh-1 appears to combine favorable recovery of Coulson-Fischer points with good numerical performance.