We present error indicators for the p- and hp-version of the boundary element method for elliptic boundary value problems in two and three dimensions. Our indicators are based on local enrichments of the ansatz spaces and decompositions thereof. Key points in the analysis of reliability are a saturation assumption and stability of the subspace decompositions. For efficiency, on the other hand, just stability of the decompositions is needed. We present various types of decompositions, in two and three space dimensions, and prove stability. For model problems, the performance of corresponding algorithms is shown and the influence of saturation on reliability is indicated.