This paper addresses finite-alphabet block-sparse signal recovery by considering support detection and data estimation separately. To this aim, we propose a maximum a posteriori (MAP) support detection criterion that takes into account the finite alphabet of the signal as a constraint. We then incorporate the MAP criterion in a compressed sensing detector based on a greedy algorithm for support estimation. We also propose to consider the finite-alphabet property of the signal in the bound-constrained least-squares optimization algorithm for data estimation. The MAP support detection criterion is investigated in two different contexts: independent linear modulation symbols and dependent binary Continous Phase Modulation (CPM) symbols. Simulations are carried out in the context of sporadic multiuser communications and show the efficiency of proposed algorithms compared to selected state-of-the-art algorithms both in terms of support detection and data estimation.