We consider a problem of blind source separation from a set of instan taneous linear mixtures, where the mixing matrix is unknown. It was discovered recently, that exploiting the sparsity of sources in an appro priate representation according to some signal dictionary, dramatically improves the quality of separation. In this work we use the property of multiscale transforms, such as wavelet or wavelet packets, to decompose signals into sets of local features with various degrees of sparsity. We use this intrinsic property for selecting the best (most sparse) subsets of features for further separation. The performance of the algorithm is verified on noise-free and noisy data. Experiments with simulated signals, musical sounds and images demonstrate significant improvement of separation quality over previously reported results.