Experiments were performed to investigate the operational modes in a dual-capillary electrospraying (ES) system in the point-to-plate configuration. For a fixed distance from the capillary tip to the plate, four possible primary modes were observed with increasing of applied voltage. They are the dripping mode, silver-bullet-like mode, stable compound cone-jet mode (if attainable), and unstable cone-jet mode (or called multi-jet mode). The characteristics (i.e., the transformation of liquid meniscus shapes and spray current) of these modes and the transitions between the modes were described. Special attention was paid to identifying the sufficient conditions to establish the stable compound cone-jet mode in the dual-capillary ES system. With the designed experimental matrix for test liquid pairs (i.e., for inner and outer liquids) it was found that a stable compound cone-jet mode can be formed for miscible and partially miscible liquid pairs. For immiscible liquid pairs, two sufficient conditions were identified for the formation of stable compound cone-jet mode: (1) the liquid of low dielectric constant in the liquid pair should be used as outer liquid, and (2) the surface tensions of the liquid pair should satisfy the total or particle engulfing conditions for a three-phase interaction system at static condition. The reasoning for the two observed sufficient conditions is also given.