  We investigate all potentially viable scenarios that can produce the chiral
enhancement required to simultaneously explain the $(g-2)_{e}$ and
$(g-2)_{\mu}$ data with either a single scalar leptoquark or a pair of scalar
leptoquarks. We provide classification of these scenarios in terms of their
ability to satisfy the existing limits on the branching ratio for the $\mu \to
e \gamma$ process. The simultaneous explanation of the $(g-2)_{e,\mu}$
discrepancies, coupled with the current experimental data, implies that the
$(g-2)_e$ loops are exclusively due to the charm quark propagation whereas the
$(g-2)_\mu$ loops are due to the top quark propagation. The scenarios where the
$(g-2)_e$ loops are due to the top (bottom) quark propagation are, at best,
approximately nine (three) orders of magnitude away from the experimental limit
on the $\mu \to e \gamma$ branching ratio. All in all, there are only three
particular scenarios that can pass the $\mu \to e \gamma$ test and
simultaneously create large enough impact on the $(g-2)_{e,\mu}$ discrepancies
when the new physics is based on the Standard Model fermion content. These are
the $S_1$, $R_2$, and $S_1 \& S_3$ scenarios, where the first two are already
known to be phenomenologically viable candidates with respect to all other
flavor and collider data constraints. We show that the third scenario, where
the right-chiral couplings to charged leptons are due to $S_1$, the left-chiral
couplings to charged leptons are due to $S_3$, and the two leptoquarks mix
through the Standard Model Higgs field, cannot address the $(g-2)_{e}$ and
$(g-2)_{\mu}$ discrepancies at the $1\sigma$ level due to an interplay between
$K_L^0 \to e^\pm \mu^\mp$, $Z \to e^+ e^-$, and $Z \to \mu^+ \mu^-$ data
despite the ability of that scenario to avoid the $\mu \to e \gamma$ limit.
