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Low-frequency (5–30 kHz) discharge current oscillations characterize the operation of Hall thrusters and represent a valuable metric not only to monitor the thruster behavior, but also to optimize the thruster performance. Two types of low-frequency oscillation modes are commonly observed: a global “breathing” mode, associated with the propellant ionization, and a local mode, typically characterized by a lower oscillation amplitude and the appearance of azimuthal spokes. The main characteristics of discharge oscillations and the transition between the two oscillation modes vary greatly with the thruster geometry and the operating condition. In this work, we present the results of an experimental campaign carried out on a 20 kW-class thruster prototype, SITAEL’s HT20k, with an exchangeable discharge channel and a magnetic circuit. Three different channel sizes were tested over a wide range of operating conditions and magnetic fields. For each operating point, a high frequency measurement of the discharge current was performed, recording the main characteristics of the oscillations. The data collected were then processed to derive the influence coefficients of each thruster parameter on the discharge current characteristics, as well as their dispersion. Finally, this allowed us to formulate general, data-driven scaling laws for the discharge current salient features, such as oscillation amplitude and dominant frequency. The gathered insight sheds light on the physical processes involved in the thruster discharge. At the same time, the possibility to model with simple functional laws the main oscillatory mode of Hall thrusters offers a unique aid to the optimization of thruster design and the evaluation of thruster performance during life.