Strong electric fields may deform drops and induce their oscillation or motion on the substrate. Moreover, they can initiate partial discharges (PDs) because of the enhancement of the electric field in the vicinity of the three-phase contact lines. The partial discharges affect the drop spreading which can result in unusual drop shapes. In addition, the partial discharges can also deteriorate the surface properties of the substrate, e.g., of high-voltage composite insulators. In this study the occurrence of partial discharges due to stationary or oscillating sessile drops under the influence of an alternating electric field is investigated using a generic insulator model under well-defined conditions. Drops of a yield stress fluid (a gelatin-water mixture) are used to determine the PD inception field strength for stationary drop shapes. The influence of the volume as well as the distance between the individual drops for two drop configurations on the PD inception threshold is determined. The inception field strength of the partial discharges is measured for various drop volumes, drop charges, as well as for different resonance modes of drop oscillations. Besides the electrical measurement, the location of the partial discharges is optically determined by a UV camera. The detailed knowledge of the influencing factors of the partial discharges improves the understanding of the drop behavior under the impact of strong electric fields.