This article presents a statistical framework to analyse brain functional Magnetic Resonance Imaging (fMRI) data. A particular emphasis is made on spatial correlation, which, contrary to the usual preprocessing step of spatial smoothing, is now part of the probabilistic model. The characterisation of regionally specific effects is done via the General Linear Model (GLM) using Posterior Probability Maps (PPMs). The spatial regularisation is defined over regression coefficients by specifying a spatial prior using Sparse Spatial Basis Functions (SSBFs), such as Wavelets. These are embedded in a hierarchical probabilistic model which, when inverted, automatically selects an appropriate subset of basis functions. The inversion of the model is done using Variational Bayes. We present results on synthetic data and on data from an event-related fMRI experiment. We conclude that SSBFs allow for spatial variations in signal smoothness, provide an increased sensitivity and are more computationally efficient than previously presented work.