Kelvin showed that a two-dimensional vortex under a two-dimensional disturbance in incompressible flow responds at a discrete set of eigenvalues, which were found by Broadbent & Moore (Phil. Trans. R. Soc. Lond. A 290, 353-371 (1979) to become unstable in a compressible fluid. It is now shown that three-dimensional perturbations are also unstable provided the wavelength is greater than some critical value that depends on the Mach number of the vortex. A critical boundary dividing stable from unstable modes is defined. Most of the results relate to a Rankine vortex, as in the previous work mentioned above, but some results are also given for a vortex with a different velocity profile within the core; qualitatively the same kind of behaviour is found.