We discuss the possibility of discriminating between different supersymmetric see-saw models by improving the experimental sensitivity to charged lepton flavour violating processes. Assuming a hierarchical neutrino mass spectrum, we classify realistic see-saw models according to how the hierarchy Delta m^2_{sun} << Delta m^2_{atm} is generated, and study the predictions of each class for the branching ratios of tau -> mu gamma and mu -> e gamma. The process tau -> mu gamma is found to be a particularly promising tool to probe the fundamental see-saw parameters, and especially to identify the origin of the large atmospheric mixing angle. Predictions for mu -> e gamma are more model-dependent. We point out that, even with an improvement of the experimental sensitivities by three orders of magnitude, both tau -> mu gamma and mu -> e gamma could escape detection in models where Delta m^2_{atm} is determined by one of the lightest right-handed neutrinos.