We discuss the phase transition and critical exponents in the random allocation model (urn model) for different statistical ensembles. We provide a unified presentation of the statistical properties of the model in the thermodynamic limit, uncover relationships between the thermodynamic potentials, and fill some lacunae in previous results on the singularities of these potentials at the critical point and behavior in the thermodynamic limit. The presentation is intended to be self-contained, so we carefully derive all formulas step by step throughout. Additionally, we comment on a quasiprobabilistic normalization of configuration weights, which was considered in some recent studies.