  Contact-drop dispensing is central to many small-scale applications, such as
direct-scanning probe lithography and micromachined fountain-pen techniques.
Accurate and controllable dispensing required for nanometer-resolved surface
patterning hinges on the stability and breakup of liquid bridges. Here, we
analytically study the stability of catenoids pinned at one contact line with
the other free to move on a substrate subject to axisymmetric and
non-axisymmetric perturbations. We apply a variational formulation to derive
the corresponding stability criteria. The maximal stability region and
stability region are represented in the favourable and canonical phase
diagrams, providing a complete description of catenoid equilibrium and
stability. All catenoids are stable with respect to non-axisymmetric
perturbations. For a fixed contact angle, there exists a critical volume below
which catenoids are unstable to axisymmetric perturbations. Equilibrium
solution multiplicity is discussed in detail, and we elucidate how geometrical
symmetry is reflected in the maximal stability and stability regions.
