Lepton Flavor Violating (LFV) observables involving charged lepton decays are strongly constrained by the existing measurements. The forthcoming experiments will improve the precision even further, requiring equally precise and efficient calculations of these processes in New Physics models. In particular, LFV observables within the Minimal Supersymmetric Standard Model (MSSM) has been recently recalculated in ref. [1], using a new technique of a purely algebraic Mass Insertion expansion of the amplitudes of ref. [2,3]. In this talk we present the most important results of this analysis. We list the updated bounds on the lepton flavor changing parameters of the MSSM derived from the measurements of the radiative decays $\ell \to \ell' \gamma$ and the 3-body decays $\ell \to 3 \ell'$ and $\ell \to 2\ell'\ell''$. We also discuss the complementarity between the radiative and 3-body decays useful for eliminating "blind spots" in the parameter space and the non-decoupling effects in the LFV Higgs decays $h \to \ell\ell'$.