Spherical tokamaks are known to differ from conventional tokamaks in a number of physics areas, but our understanding is limited by a comparative lack of experimental data. A comprehensive study of the density dependence of the H-mode power threshold P_LH on the Mega-Amp Spherical Tokamak (MAST) is presented, mapping out the low-density and high-density branches, and describing different types of L-H transitions and intermediate behaviours. Transitions at low densities are characterised by longer preceding 3-4kHz i-phase oscillations or distinct dithery periods. Commonly used P_LH scalings underestimate the experimental values by at least an order of magnitude, and a fit to the high-density branch gives P_LH=11.35×n_e20^1.19, which has a higher density exponent than the scalings. One mechanism thought to be of importance in the development of a shear in the radial electric field during an L-H transition is the ion heat flux in the edge. Transitions of different densities and neutral beam heating powers were analysed with the interpretative transport code TRANSP, revealing that the total heat flux Q_tot possessed a density dependence independent of the net power. The density dependence of the electron heat flux Q_e~0.62n_e is stronger than that of the ion heat flux Q_i~0.15n_e, with Q_i contributing ~12-30% of Q_tot. The density dependence is caused by a decrease in beam heating efficiency for both lower densities and for higher injected powers P_inj with the fraction of P_inj heating the plasma decreasing from 80% at high densities to 30% at low densities. Low-density or high-P_inj discharges appear to have greater fast ion losses, majority orbit and charge-exchange losses, which are seen in Mirnov signals as chirping 50-20kHz modes and broadband MHD at 150-250kHz. If the P_LH study is adjusted to account for fast ion losses, the scatter in H-mode points is reduced, the density exponent is increased to P_LH=11.27×n_e^1.33, but low-density dithery transitions still lie above the P_LH curve, so the low-density branch is not entirely explained.