In this paper, we focus on modeling multichannel audio signals in the short-time Fourier transform domain for the purpose of source separation. We propose a probabilistic model based on a class of heavy-tailed distributions, in which the observed mixtures and the latent sources are jointly modeled by using a certain class of multivariate alpha-stable distributions. As opposed to the conventional Gaussian models, where the observations are constrained to lie just within a few standard deviations near the mean, the pro- posed heavy-tailed model allows us to account for spurious data or important uncertainties in the model. We develop a Monte Carlo Expectation-Maximization algorithm for making inference in the proposed model. We show that our approach leads to significant improvements in audio source separation under corrupted mixtures and in spatial audio object coding.