The stability of an equilibrium system of two drops suspended from circular holes in a horizontal plate is examined. The drop surfaces are the disconnected axisymmetric surfaces pinned to the edges of the holes. The holes lie in the same horizontal plane and the two drops are connected by a liquid layer that lies above the plate. The total liquid volume is constant. For identical pendant drops pinned to holes of equal radii, axisymmetric perturbations are always the most dangerous. The stability region for two identical drops differs considerably from that for a solitary pendant drop. A bifurcation analysis shows that the loss of stability leads to a continuous transition from a critical system of identical drops to a stable system of axisymmetric non-identical drops. With increasing total protruded liquid volume this system of non-identical drops reaches its own collective stability limit (to axisymmetric perturbations) which gives rise to dripping or streaming from the holes. Critical volumes and heights for non-identical drops have been calculated as functions of the dimensionless hole radius (associated with the Bond number). For unequal hole radii, there are three intervals of the larger dimensionless hole radius, R10, with qualitatively different bifurcation patterns which in turn can depend on the smaller dimensionless hole radius, R10. Loss of stability may occur when the drop suspended from the larger hole reaches its stability limit (to non-axisymmetric perturbations) as a solitary drop or when the system reaches the collective stability limit (to axisymmetric perturbations). Typical situations are illustrated for selected values of R10, and then the basic characteristics of the stability for a dense set of R10 are presented. © 2006 Cambridge University Press.