The inviscid interaction of vortex structures has been studied extensively to elucidate many of the important features of high Reynolds number flows and turbulence. Contour dynamics was developed specifically to study the evolution of the boundaries of uniform vortices, and has been used to study a wide variety of flows. The method is a lagrangian one, and typically a discrete dynamical system is obtained by representing the boundary by a finite collection of points. We examine the linear stability features of a typical method by considering perturbations to a circular, uniform vortex. We find a spurious branch to the numerical dispersion relation that indicates points will oscillate along the boundary without any distortion in its shape. Numerical results for the full equations show that the amplitudes of these particular modes grow slowly in time.