Electronic-vibrational coupling in single-molecule junctions may result in current-induced bond rupture and is thus an important mechanism for the stability of molecular junctions. We use the hierarchical quantum master equation method in combination with the quasiclassical Ehrenfest approach for the nuclear degrees of freedom to simulate current-induced bond rupture in single-molecule junctions. Employing generic models for molecular junctions with dissociative nuclear potentials, we analyze the underlying mechanisms. In particular, we investigate the dependence of the dissociation probability on the applied bias voltage and the molecule-lead coupling strength. The results show that an applied bias voltage can not only lead to dissociation of the molecular junction, but under certain conditions can also increase the stability of the molecule.