Résumé
Upcoming experiments will improve the sensitivity to $\mu\to e$ processes by several orders of magnitude, and could observe lepton flavour-changing contact interactions for the first time. In this paper, we investigate what could be learned about New Physics from the measurements of these $\mu\to e$ observables, using a bottom-up effective field theory (EFT) approach and focusing on three popular models with new particles around the TeV scale (the type II seesaw, the inverse seesaw and a scalar leptoquark). We showed in a previous publication that $\mu\to e$ observables have the ability to rule out these models because none can fill the whole experimentally accessible parameter space. In this work, we give more details on our EFT formalism and present more complete results. We discuss the impact of some observables complementary to $\mu\to e$ transitions (such as the neutrino mass scale and ordering, and LFV $\tau$ decays) and draw attention to the interesting appearance of Jarlskog-like invariants in our expressions for the low-energy Wilson coefficients.