  We study a new supersymmetric mechanism for lepton flavour violation in muon
and tau decays and muon -> e conversion in nuclei, within a minimal extension
of the MSSM with low-mass heavy singlet neutrinos and sneutrinos. We find that
the decays mu -> e gamma, tau -> e gamma and tau -> mu gamma are forbidden in
the supersymmetric limit of the theory, whereas other processes, such as mu ->
eee, mu -> e conversion, tau -> eee and tau -> e mu mu, are allowed and can be
dramatically enhanced several orders of magnitude above the observable level by
potentially large neutrino Yukawa coupling effects. The profound implications
of supersymmetric lepton flavour violation for present and future experiments
are discussed.
