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The strong statistical significance of an observed electron-like event excess in the MiniBooNE (MB) experiment, along with an earlier similar excess seen in the Liquid Scintillator Neutrino Detector (LSND), when interpreted in conjunction with recent MicroBooNE results may have brought us to the cusp of new physics discoveries. This has led to many attempts to understand these observations, both for each experiment individually and in conjunction, via physics beyond the Standard Model (SM). We provide an overview of the current situation, and discuss three major categories under which the many proposals for new physics fall. The possibility that the same new, non-oscillation physics explains both anomalies leads to new restrictions and requirements. An important class of such common solutions, which we focus on in this work, consists of a heavy ${\cal O}$(MeV$-$sub-GeV) sterile neutral fermion produced in the detectors, (via up-scattering of the incoming muon neutrinos), and subsequently decaying to photons or $e^+e^-$ pairs which mimic the observed signals. Such solutions are subject to strong demands from a) cross section requirements which would yield a sufficient number of total events in both LSND and MB, b) requirements imposed by the measured energy and angular distributions in both experiments and finally, c) consistency and compatibility of the new physics model and its particle content with other bounds from a diverse swathe of particle physics experiments. We find that these criteria often pull proposed solutions in different directions, and stringently limit the viable set of proposals which could resolve both anomalies. Our conclusions are relevant for both the general search for new physics and for the ongoing observations and analyses of the MicroBooNE experiment.