We present the search strategies for two different leptoquark models at the present and future colliders. A model containing a singlet and a doublet scalar leptoquark can generate neutrino mass at one-loop while contributing to the muon $g-2$ experimental measurement. Signatures of this model, in benchmark scenarios that simultaneously satisfy neutrino mass and oscillation data, muon $g-2$ excess as well as CLFV bounds, are studied from the pair production at the LHC/FCC, with complementary final states that distinguish the leptoquark mass eigenstates. A non-trivial mixing exists between the singlet and the doublet, which can be probed by an asymmetric pair production. Next, we consider a singlet and a triplet scalar leptoquark separately, motivated by their contribution to $B$-decay ratios and associated observables. A benchmark-independent study is performed for the 5$\sigma$ probe of a very minimal set of Yukawa couplings at the LHC/FCC from the single production of leptoquarks. A complementary study from the pair production is performed at a multi-TeV muon collider, which leads to cleaner signal and better reach.