Disconnected free surfaces (or interfaces) of a connected liquid volume (or liquid volumes) occur when the boundary of the liquid volume consists of two or more separate surface components Γi (i = 1,...,m) that correspond to liquid-gas (or liquid-liquid) interfaces. We consider disconnected surfaces for which each component Γi is axisymmetric and crosses its own symmetry axis. In most cases, the stability problem for an entire disconnected equilibrium capillary surface subject to perturbations that conserve the total liquid volume reduces to the same set of problems obtained when separately considering the stability of each Γi to perturbations that satisfy a fixed pressure constraint. For fixed pressure perturbations, the stability of a given axisymmetric Γi can be found through comparison of actual and critical values of a particular boundary parameter. For zero gravity, these critical values are found analytically. For non-zero gravity, an analytical representation of the critical values is not generally possible. In such cases, a determination of stability can be accomplished by representing all possible equilibrium surface profiles on a dimensionless "height-Dadius" diagram. This diagram is contoured with critical values of the boundary parameter. The stability diagram can, in most cases, be used to determine the stability of a disconnected surface (subject to perturbations that conserve the total volume) that is composed of components that are represented by given equilibrium profiles on the diagram. To illustrate this approach, solutions of stability problems for systems consisting of a set of sessile or pendant drops in contact with smooth planar walls or with the edges of equidimensional perforated holes in a horizontal plate are presented. © 2003 American Institute of Physics.