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This paper proposes a new Bayesian nonnegative matrix factorization (NMF) for speech and music separation. We introduce the Poisson likelihood for NMF approximation and the exponential prior distributions for the factorized basis matrix and weight matrix. A variational Bayesian (VB) EM algorithm is developed to implement an efficient solution to variational parameters and model parameters for Bayesian NMF. Importantly, the exponential prior parameter is used to control the sparseness in basis representation. The variational lower bound in VB-EM procedure is derived as an objective to conduct adaptive basis selection for different mixed signals. The experiments on single-channel speech/music separation show that the adaptive basis representation in Bayesian NMF via model selection performs better than the NMF with the fixed number of bases in terms of signal-to-distortion ratio.