  Supersymmetry with R-parity violation (RPV) provides an interesting framework
for naturally accommodating small neutrino masses. Within this framework, we
discuss the lepton-flavour violating (LFV) processes mu -> e gamma, mu -> eee,
and mu -> e conversion in nuclei. We make a detailed study of the observables
related to LFV in different RPV models, and compare them to the expectations of
R-conserving supersymmetry with heavy right-handed neutrinos. We show that the
predictions are vastly different and uniquely characterise each model, thus
providing a powerful framework for experimentally distinguishing between
different theories of LFV. Besides the obvious possibility of amplified
tree-level generation of mu -> eee and mu -> e conversion in nuclei, we find
that even in the case where these processes arise at the one-loop level, their
rates are comparable to that of mu -> e gamma, in clear contrast to the
predictions of R-conserving models. We conclude that in order to distinguish
between the different models, such a combined study of all the LFV processes is
necessary, and that measuring P-odd asymmetries in polarised mu -> eee can play
a decisive role. We also comment on the intriguing possibility of RPV models
yielding a large T-odd asymmetry in the decay of polarised mu -> eee.
