  Lepton flavour non-conserving processes are examined in the context of
unified models with U(1)-family symmetries which reproduce successfully the
low-energy hierarchy of the fermion mass spectrum and the Kobayashi - Maskawa
mixing. These models usually imply mixing effects in the supersymmetric scalar
sector. We construct the fermion and scalar mass matrices in two viable models,
and calculate the mixing effects on the $\mu \to e\gamma$, $\mu \to 3 e$ and
$\tau \to \mu\gamma$ rare decays. The relevant constraints on the sparticle
mass spectrum as well as the role of various MSSM parameters are discussed.
