  We present a comprehensive study of two single-reference approaches to
singlet biradicaloids. These two approaches are based on the recently developed
regularized orbital-optimized M{\o}ller-Plesset method ($\kappa$-OOMP2). The
first approach is to combine the Yamaguchi's approximate projection (AP) scheme
and $\kappa$-OOMP2 with unrestricted (U) orbitals ($\kappa$-UOOMP2). By
capturing only essential symmetry breaking, $\kappa$-UOOMP2 can serve as a
suitable basis for AP. The second approach is $\kappa$-OOMP2 with complex,
restricted (cR) orbitals ($\kappa$-cROOMP2). Though its applicability is more
limited due to the comparative rarity of cR solutions, $\kappa$-cROOMP2 offers
a simple framework for describing singlet biradicaloids with complex
polarization while removing artificial spatial symmetry breaking. We compare
the scope of these two methods with numerical studies. We show that
AP+$\kappa$-UOOMP2 and $\kappa$-cROOMP2 can perform similarly well in the TS12
set, a data set that includes 12 data points for triplet-singlet gaps of
several atoms and diatomic molecules with a triplet ground state. This was also
found to be true for the barrier height of a reaction involving attack on a
cysteine ion by a singlet oxygen molecule. However, we also demonstrate that in
highly symmetric systems like $\text{C}_{30}$ ($\text{D}_{5h}$)
$\kappa$-cROOMP2 is more suitable as it conserves spatial symmetry. Lastly, we
present an organic biradicaloid that does not have a $\kappa$-cROOMP2 solution
in which case only AP+$\kappa$-UOOMP2 is applicable. We recommend
$\kappa$-cROOMP2 whenever complex polarization is essential and
AP+$\kappa$-UOOMP2 for biradicaloids without essential complex polarization but
with essential spin-polarization.
