A detailed investigation has been made of the transverse sawtooth instability observed at the Advanced Photon Source. This horizontal single-bunch instability exhibits two modes: steady-state, with continuous centroid oscillations at fixed amplitude, and bursting, with an amplitude that varies quasi-periodically with a peak-topeak excursion of up to 2 mm. The transition between modes is sharp and, depending on machine conditions, multiple transitions are observed with increasing bunch intensity. The threshold and character of the instability also vary with chromaticity and rf voltage. The horizontal instability shows striking similarities to the longitudinal sawtooth instability observed in the Stanford Linear Collider damping ring, as well as clear and obvious differences. This paper presents experimental results and a possible mechanism for the sawtooth instability that is suggested by simulations.