This paper introduces a novel framework for compressive sensing of biomedical ultrasonic signals based on modelling data with stable distributions. We propose an approach to ℓ(p) norm minimisation that employs the iteratively reweighted least squares (IRLS) algorithm but in which the parameter p is judiciously chosen by relating it to the characteristic exponent of the underlying alpha-stable distributed data. Our results show that the proposed algorithm, which we prefer to call S ± S-IRLS, outperforms previously proposed ℓ(1) minimisation algorithms, such as basis pursuit or orthogonal matching pursuit, both visually and in terms of PSNR.