The stability of vortex rings with an azimuthal component of velocity is investigated numerically for various combinations of ring wavenumber and swirl magnitude. The vortex rings are equilibrated from an initially Gaussian distribution of azimuthal vorticity and azimuthal velocity, at a circulation-based Reynolds number of 10 000, to a state in which the vortex core is qualitatively identical to that of the piston generated vortex rings. The instability modes of these rings can be characterised as Kelvin instability modes, analogous to instability modes observed for Gaussian and Batchelor vortex pairs. The shape of an amplified mode typically depends only on the azimuthal wavenumber at the centre of the vortex core and the magnitude of the corresponding velocity component. The wavenumber of a particular sinuous instability varies with radius from the vortex ring centre for rings of finite aspect ratio. Thicker rings spread the amplification over a wider range of wavenumbers for a particular resonant mode pair, while the growth rate and the azimuthal wavenumber corresponding to the peak growth both vary as a function of the wavenumber variation. Normalisation of the wavenumber and the growth rate by a measure of the wavenumber variation allows a coherent description of stability modes to be proposed, across the parameter space. These results provide a framework for predicting the development of resonant Kelvin instabilities on vortex rings with an induced component of swirling velocity.