  The forthcoming precision data on lepton flavour violating decays require
precise and efficient calculations in New Physics models. In this article
lepton flavour violating processes within the Minimal Supersymmetric Standard
Model (MSSM) are calculated using the method based on the Flavour Expansion
Theorem, a recently developed technique performing a purely algebraic
mass-insertion expansion of the amplitudes. The expansion in both
flavour-violating and flavour-conserving off-diagonal terms of sfermion and
supersymmetric fermion mass matrices is considered. In this way the relevant
processes are expressed directly in terms of the parameters of the MSSM
Lagrangian. We also study the decoupling properties of the amplitudes. The
results are compared to the corresponding calculations in the mass eigenbasis
(i.e. using the exact diagonalization of the mass matrices). Using these
methods, we consider the following processes: $\ell \to \ell' \gamma$, $\ell
\to 3 \ell'$, $\ell \to 2\ell'\ell"$, $h \to \ell\ell'$ as well as $\mu \to e$
conversion in nuclei. In the numerical analysis we update the bounds on the
flavour changing parameters of the MSSM and examine the sensitivity to the
forthcoming experimental results. We find that flavour violating muon decays
provide the most stringent bounds on supersymmetric effects and will continue
to do so in the future. Radiative $\ell\to\ell^\prime\gamma$ decays and
leptonic three-body decays $\ell\to 3\ell^\prime$ show an interesting
complementarity in eliminating "blind spots" in the parameter space. In our
analysis we also include the effects of non-holomorphic $A$-terms which are
important for the study of LFV Higgs decays.
