The instability of a vortex ring is investigated numerically by a three-dimensional vortex method. Disturbances with small amplitude are imposed initially over a local region of a vortex ring, and the development of large-scale instability waves and/or small-scale irregular motions of vortices along the vortex core is studied. Three different types of disturbances are considered : (1) sinusoidal disturbances with wave number m=6 (hexagonal), (2) random disturbances, and (3) random disturbances superimposed on sinusoidal (m=6) disturbances. It is found that locally imposed sinusoidal disturbances propagate along a vortex core and produce large-scale instability waves over the whole region of a vortex ring. On the other hand, the propagation velocity of locally imposed random disturbances along a vortex core is negligibly small, and the development of small-scale irregular motions of vortices is almost confined within the local region.