  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.
