Our measured dielectric constant and mechanical response of multiferroic BiFeO3 indicate four phase transitions below room temperature. Features correlate with those reported at 50K (from a peak in the zero-field-cooled magnetic susceptibility) and 230K (from splitting between field-cooled and zero-field-cooled magnetic data[1], and 200K (from magnon light scattering cross sections[2]). The primary order parameter is not the polarization in any of the low-T transitions. Instead, the transition near 230 K shows strong elastic coupling, while that at 50K is fundamentally magnetic, but magnetostrictively coupled to the the lattice. The low-T phase transitions display glassy behaviour. A further anomaly at 140K interpreted as spin reorientation[2,3] shows only weakly in dielectric and mechanical studies, indicating that it is predominantly magnetic with little coupling to any of the other order parameters.