  We study the rates allowed for the Higgs-mediated decays
$B_{s,d}^0\to\mu\tau, e\tau$ and $\tau\to \mu\mu\mu, e\mu\mu$ in supersymmetric
seesaw models, assuming that the only source of lepton flavour violation (LFV)
is the renormalization of soft supersymmetry-breaking terms due to off-diagonal
singlet-neutrino Yukawa interactions. These decays are strongly correlated
with, and constrained by, the branching ratios for $B_{s,d}^0\to\mu\mu$ and
$\tau\to \mu(e)\gamma.$ Parametrizing the singlet-neutrino Yukawa couplings
$Y_\nu$ and masses $M_{N_i}$ in terms of low-energy neutrino data, and allowing
the flavour-universal soft masses for sleptons and for squarks, as well as
those for the two Higgs doublets, to be different at the unification scale, we
scan systematically over the model parameter space. Neutrino data and the
present experimental constraints set upper limits on the Higgs-mediated LFV
decay rates $Br(B_{s}^0\to\mu\tau, e\tau)\lsim 4\times 10^{-9}$ and
$Br(\tau\to\mu\mu\mu, e\mu\mu)\lsim 4\times 10^{-10}$.
