The forthcoming precision data on lepton flavour violating (LFV) 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: ℓ → ℓ ′ γ, ℓ→3ℓ ′ , ℓ→2ℓ ′ ℓ ′′ , h→ℓℓ ′ as well as μ→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. ... mehrWe find that flavour violating muon decays provide the most stringent bounds on supersymmetric effects and will continue to do so in the future. Radiative ℓ → ℓ ′ γ decays and leptonic three-body decays ℓ → 3ℓ ′ 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.