  We compare dynamical and energetical stability criteria for vortex rings. It
is argued that vortex rings will be intrinsically unstable against
perturbations with short wavelengths below a critical wavelength, because the
canonical vortex Hamiltonian is unbounded from below for these modes. To
explicitly demonstrate this behaviour, we derive the oscillation spectrum of
vortex rings in incompressible, inviscid fluids, within a geometrical cutoff
procedure for the core. The spectrum develops an anomalous branch of negative
group velocity, and approaches the zero of energy for wavelengths which are
about six times the core diameter. We show the consequences of this dispersion
relation for the thermodynamics of vortex rings in superfluid $^4$He at low
temperatures.
