Most methods for sparse signal recovery require to set one or several hyperparameters. We propose an unsupervised method to estimate the parameters of a Bernoulli-Gaussian (BG) model describing sparse signals. The proposed method is first derived for denoising problems, based on a maximum likelihood (ML) approach. Then, an extention to general inverse problems is achieved through a latent variable formulation. Two expectation- maximization (EM) algorithms are then proposed to estimate the signal together with the BG model parameters. Combining these two approaches leads to the proposed LEMUR algorithm. All proposed algorithms are then evaluated on extensive simulations in terms of ability to recover the parameters and provide accurate sparse signal estimates.