The aim of spectral unmixing of hyperspectral images is to determine the component materials and their associated abundances from mixed pixels. In this paper, we present sparse linear unmixing via an Expectation-Propagation method based on the classical linear mixing model and a spike-and-slab prior promoting abundance sparsity. The proposed method, which allows approximate uncertainty quantification (UQ), is compared to existing sparse unmixing methods, including Monte Carlo strategies traditionally considered for UQ. Experimental results on synthetic data and real hyperspectral data illustrate the benefits of the proposed algorithm over state-of-art linear unmixing methods.