The simplest unified extension of the Minimal Supersymmetric Standard Model with bi-linear R-parity violation naturally predicts a hierarchical neutrino mass spectrum, suitable to explain atmospheric and solar neutrino fluxes. We study whether the individual violation of the lepton numbers L e,µ,τ in the charged sector can lead to measurable rates for BR(µ → eγ) and BR(τ → µγ). We find that some of the R-parity violating terms that are compatible with the observed atmospheric neutrino oscillations could lead to rates for µ → eγ measurable in projected experiments. However, the ∆m 2 12 obtained for those parameters is too high to be compatible with the solar neutrino data, excluding therefore the possibility of having measurable rates for µ → eγ in the model.