Sparse regression often uses $\ell_p$ norm priors (with $p<2$). This paper demonstrates that the introduction of mixed-norms in such contexts allows one to go one step beyond in signal models, and promote some different, structured, forms of sparsity. It is shown that the particular case of $\ell_{1,2}$ and $\ell_{2,1}$ norms lead to new group shrinkage operators. Two different problems are considered, that illustrate the relevance of the proposed approach, in the context of audio signals. Mixed norm priors are shown to be particularly efficient for multichannel audio denoising, in a generalized basis pursuit denoising approach. Mixed norm priors are also used in a context of morphological component analysis of time-frequency sound representations, for which an adapted version of Block Coordinate Relaxation algorithm is derived. This yields a new approach for sparse regression in time-frequency dictionaries.