The advent of new combinations of equipment and integrations of different power grids can potentially be accompanied by transient voltage phenomena which affect the reliability of electrical insulation systems. In particular, this affects insulation susceptible to partial discharge behaviour due to additional transient waveforms. Nowadays in transmission grid planning, overhead lines are increasingly combined with underground power cables. In switching such connections impulses could occasionally occur and superimpose on the normal AC voltage which will lead to superimposed transient waveforms. Those transient voltages will impose an additional stress on the cable insulation that could trigger partial discharges, that sustain. This paper reviews our recent research of some cable insulation defects of industrial relevance by partial discharge measurement on high voltage power cables stressed by different waveforms, including impulses, superimposed transients on AC voltage. This paper identifies achievements and at the same time challenges still to be solved in partial discharge measurement under different waveforms.