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We investigate all potentially viable scenarios that can produce the chiralenhancement required to simultaneously explain the $(g-2)_{e}$ and$(g-2)_{\mu}$ data with either a single scalar leptoquark or a pair of scalarleptoquarks. We provide classification of these scenarios in terms of theirability to satisfy the existing limits on the branching ratio for the $\mu \toe \gamma$ 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 limiton the $\mu \to e \gamma$ branching ratio. All in all, there are only threeparticular scenarios that can pass the $\mu \to e \gamma$ test andsimultaneously create large enough impact on the $(g-2)_{e,\mu}$ discrepancieswhen the new physics is based on the Standard Model fermion content. These arethe $S_1$, $R_2$, and $S_1 \& S_3$ scenarios, where the first two are alreadyknown to be phenomenologically viable candidates with respect to all otherflavor and collider data constraints. We show that the third scenario, wherethe right-chiral couplings to charged leptons are due to $S_1$, the left-chiralcouplings to charged leptons are due to $S_3$, and the two leptoquarks mixthrough 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 $Z \to \mu^+ \mu^-$ datadespite the ability of that scenario to avoid the $\mu \to e \gamma$ limit.

$\mu \to e \gamma$ selecting scalar leptoquark solutions for the $(g-2)_{e,\mu}$ puzzles