The bifurcation structure of thermohaline-driven flows is studied within one of the simplest zonally averaged models which captures thermohaline transport: a Boussinesq model of surface-forced thermohaline flow in a two-dimensional rectangular basin. Under mixed boundary conditions, i.e. prescribed surface temperature and fresh-water flux, it is shown that symmetry breaking originates from a codimension-two singularity which arises through the intersection of the paths of two symmetry-breaking pitchfork bifurcations. The physical mechanism of symmetry breaking of both the thermally and salinity dominated symmetric solution is described in detail from the perturbation structures near bifurcation. Limit cycles with an oscillation period in the order of the overturning time scale arise through Hopf bifurcations on the branches of asymmetric steady solutions. The physical mechanism of oscillation is described in terms of the most unstable mode just at the Hopf bifurcation. The occurrence of these oscillations is quite sensitive to the shape of the prescribed fresh-water flux. Symmetry breaking still occurs when, instead of a fixed temperature, a Newtonian cooling condition is prescribed at the surface. There is only quantitative sensitivity, i.e. the positions of the bifurcation points shift with the surface heat transfer coefficient. There are no qualitative changes in the bifurcation diagram except in the limit where both the surface heat flux and fresh-water flux are prescribed. The bifurcation structure at large aspect ratio is shown to converge to that obtained by asymptotic theory. The complete structure of symmetric and asymmetric multiple equilibria is shown to originate from a codimension-three bifurcation, which arises through the intersection of a cusp and the codimension-two singularity responsible for symmetry breaking.