The degradation dynamics and post-breakdown current-voltage (I-V) characteristics of magnesium oxide (MgO) layers grown on n and p-type indium phosphide (InP) substrates subjected to electrical stress were investigated. We show that the current-time (I-t) characteristics during degradation can be described by a power-law model I(t) - I(0)t(-alpha). where I-0 and alpha are constants. It is reported that the leakage current associated with the soft breakdown (SBD) failure mode follows the typical voltage dependence I = aV(b), where a and b are constants, for both injection polarities but in a wider voltage range compared with the SiO2/Si system. It is also shown that the hard breakdown (HBD) current is remarkably high, involving large ON-OFF fluctuations that resemble the phenomenon of resistive switching previously observed in a wide variety of metal oxides.