This paper describes a new method for lossy audio signal compression via compressive sensing (CS). In this method, a structured shrinkage operator is employed to decompose the audio signal into three layers, with two sparse layers, tonal and transient, and additive noise, and then, both the tonal and transient layers are compressed using CS. Since the shrinkage operator is able to take into account the structure information of the coefficients in the transform domain, it is able to achieve a better sparse approximation of the audio signal than traditional methods do. In addition, we propose a sparsity allocation algorithm, which adjusts the sparsity between the two layers, thus improving the performance of CS. Experimental results demonstrated that the new method provided a better compression performance than conventional methods did.