The Brillouin flow is the dominant flow in crossed-field devices. We systematically study its stability in conventional, planar, and inverted magnetron. We find that the negative mass effect in the inverted magnetron tends to destabilize the planar flow, and that the positive mass effect in the conventional magnetron tends to stabilize the planar flow. We find that the presence of a slow-wave structure, which is customarily ignored in prior studies, greatly changes the stability of the Brillouin flow.