Biomechanical modeling and bifurcation theory are applied to study phonation onset and register transition. A four-mass body-cover model with a smooth geometry is introduced to reproduce characteristic features of chest and falsetto registers. Sub- and supraglottal resonances are modeled using a wave-reflection model. Simulations for increasing and decreasing subglottal pressure reveal that the phonation onset exhibits amplitude jumps and hysteresis referring to a subcritical Hopf bifurcation. The onset pressure is reduced due to vocal tract resonances. Hysteresis is observed also for the voice breaks at the chest-falsetto transition. Varying the length of the subglottal resonator has only minor effects on this register transition. Contrarily, supraglottal resonances have a strong effect on the pitch, at which the chest-falsetto transition is found. Experiment of glissando singing shows that the supraglottis has indeed an influence on the register transition.