The soft breakdown (SBD) failure mode in 20 nm thick MgO dielectric layers grown on Si substrates was investigated. We show that during a constant voltage stress, charge trapping and progressive breakdown coexist, and that the degradation dynamics is captured by a power-law time dependence. We also show that the SBD current-voltage (I-V) characteristics follow the power-law model I = aVb typical of this conduction mechanism but in a wider voltage window than the one reported in the past for SiO2. The relationship between the magnitude of the current and the normalized differential conductance was analyzed.