The blind source separation problem is to extract the underlying source signals from a set of their linear mixtures, where the mixing matrix is unknown. This situation is common, eg in acoustics, radio, and medical signal processing. We exploit the property of the sources to have a sparse representation in a corresponding (possibly overcomplete) signal dictionary. Such a dictionary may consist of wavelets, wavelet packets, etc., or be obtained by learning from a given family of signals. Starting from the maximum posteriori framework, which is applicable to the case of more sources than mixtures, we derive a few other categories of objective functions, which provide faster and more computations, when there are an equal number of sources and mixtures. Our experiments with artificial signals and with musical sounds demonstrate significantly better separation than other known techniques.