Exploiting intrinsic structures in sparse signals underpins the recent progress in compressive sensing (CS). The key is to exploit such structures to achieve the two desirable properties: generality (ie, the ability to fit a wide range of signals with diverse structures) and adaptability (ie, being adaptive to a specific signal). Most existing approaches, however, often only achieve one of these two properties. In this study, we propose a novel adaptive Markov random field sparsity prior for CS, which not only is able to support a broad range of sparsity structures but also can adapt to each sparse signal through refining the parameters of the sparsity prior with respect to the compressed measurements. To this end, the sparse signal recovery and the estimation of the parameters in the sparsity prior are jointly integrated into a unified variational optimization problem, which can be effectively solved with an alternative minimization scheme. Extensive experiments on three real-world datasets demonstrate the effectiveness of the proposed method in recovery accuracy, noise tolerance, and runtime.