New Physics can be manifest at the high intensity frontier through a vast array of phenomena, which includes rare transitions and decays (forbidden in the Standard Model), or tiny deviations from the Standard Model's expectation. Muons are very powerful and versatile probes of New Physics - numerous experiments rely on the study of ``muon channels'' to search for processes violating lepton number, charged lepton flavours, or even the universality of lepton flavours, as well as CP violation in the lepton sector. Following a short overview of the status of dedicated experimental searches, we comment on how muon observables can shed light on the underlying model of New Physics at work.