In Lipschitz two- and three-dimensional domains, we study the existence for the so-called Boussinesq model of thermally driven convection under singular forcing. By singular we mean that the heat source is allowed to belong to H-1(pi, Omega), where pi is a weight in the Muckenhoupt class A(2) that is regular near the boundary. We propose a finite element scheme and, under the assumption that the domain is convex and pi(-1) is an element of A(1), show its convergence. In the case that the thermal diffusion and viscosity are constants, we propose an a posteriori error estimator and show its reliability. We also explore efficiency estimates.