This paper concerns the problem of sparse signal recovery with multiple measurement vectors, where the sparse signal vectors share multiple supports (i.e., the signal vectors can be clustered and the vectors in a cluster share a common support) and the prior knowledge on the supports of the vectors is unknown. This problem can be solved using sparse Bayesian learning (SBL) with Dirichlet process (DP) as hyper-prior, which is named DP-SBL in this paper. This work aims to design efficient inference algorithms. The variational inference for DP mixtures, in particular mean field (MF) inference, has been studied, and applying it to the problem in this paper leads to an MF-DP-SBL algorithm. In this paper, we propose a combined message passing (CMP) approach, where a factor graph representation is designed to enable a more efficient implementation with both the MF and approximate message passing (AMP), leading to a CMP-DP-SBL algorithm. It is shown that, compared to MF-DP-SBL, CMP-DP-SBL delivers the same or even better performance with significantly lower complexity. As an example, we apply it to massive MIMO channel estimation where, due to the large number of antennas deployed at base station, the channel impulse responses measured at receive antennas can share multiple supports. It is shown that CMP-DP-SBL delivers considerably better performance than existing algorithms.