We propose a new framework, called piecewise linear separation, for blind source separation of possibly degenerate mixtures, including the extreme case of a single mixture of several sources. Its basic principle is to: 1/ decompose the observations into "components" using some sparse decomposition/nonlinear approximation technique; 2/ perform separation on each component using a "local" separation matrix. It covers many recently proposed techniques for degenerate BSS, as well as several new algorithms that we propose. We discuss two particular methods of multichannel decompositions based on the Best Basis and Matching Pursuit algorithms, as well as several methods to compute the local separation matrices (assuming the mixing matrix is known). Numerical experiments are used to compare the performance of various combinations of the decomposition and local separation methods. On the dataset used for the experiments, it seems that BB with either cosine packets of wavelet packets (Beylkin, Vaidyanathan, Battle3 or Battle 5 filter) are the best choices in terms of overall performance because they introduce a relatively low level of artefacts in the estimation of the sources; MP introduces slightly more artefacts, but can improve the rejection of the unwanted sources.