In the context of Standard Model extensions via Majorana sterile fermions, the presence of additional CP violating phases (Dirac and Majorana) has been shown to be at source of important effects in charged lepton flavour violating (cLFV) transitions and decays. Here we will consider further angular observables that can be studied for polarised $\tau$ and $\mu$ cLFV decays. These include, among others, parity asymmetries and time-reversal asymmetries for generic cLFV 3-body decays, $\ell_\alpha^+ \to \ell_\beta^+ \ell_\gamma^+ \ell_\delta^-$. We address relevant correlations between the different classes of observables, and show that one can have sizeable asymmetries, which can be used to further probe this interesting class of SM extensions. Our study leads to the prediction of particular patterns of angular observables, which would allow to potentially falsify the model, should a cLFV signal be observed.

Within the context of heavy neutral lepton (HNL) extensions of the Standard Model, we compute the cross-sections for $\mu^+ e^-\to \ell_\alpha^+\ell_\beta^-$ scattering, as well as several angular observables. In particular, we investigate the future sensitivity of a $\mu$TRISTAN collider in discovering such charged lepton flavour violating processes and the potential constraining power of these searches on the parameter space of HNL models. Our results show that while low-energy probes of $\mu-e$ flavour violation do offer the most promising potential, the prospects for $e\tau$ and $\mu\tau$ flavour violation searches at $\mu$TRISTAN can exceed those of related low-energy probes (as well as flavour violating $Z$-pole processes at FCC-ee) by several orders of magnitude.

The Inverse Seesaw mechanism remains one of the most attractive explanations for the lightness of neutrino masses, allowing for natural low-scale realisations. We consider the prospects of a simple extension via 3 generations of sterile fermions - the so called ISS(3,3) - in what concerns numerous lepton flavour observables. In order to facilitate a connection between the Lagrangian parameters and low-energy data, we systematically develop new parametrisations of the Yukawa couplings. Relying on these new parametrisations to explore the parameter space, we discuss the complementary role of charged lepton flavour violation searches in dedicated facilities, as well as in lepton colliders (FCC-ee and $\mu$TRISTAN). Our results reveal the strong synergy of the different indirect searches in probing the distinct flavour sectors of the model. In particular, we show that in the absence of radiative decays $\ell_\alpha\to\ell_\beta\gamma$, sizeable rates for $Z$-penguin dominated observables could hint at a non-trivially mixed and non-degenerate heavy spectrum.