The purpose of this paper is to investigate the performance of several adaptive wavelet constructions based on a modification of the original Lepski algorithm. First, we provide a wide class of procedures which have the particularity to attain the asymptotic minimax rate of convergence over certain zones of Besov balls under the L2 risk. Second, the method developed by Juditsky is studied. In particular, we show that it is superior to other wavelet estimators in terms of maxiset (and minimax) properties.