In this paper we present new results which illustrate the existence of trapped modes on a thin elastic plate, which is infinitely long, of finite uniform width and simply-supported along the two parallel edges. The plates contain a circular cut-out on the centreline of the strip whose edges are either free or clamped. The trapped waves describe time-harmonic vibrations of a frequency below a cut-off frequency which are localised to the circular cut-out and decay along the strip. Results show that whilst no trapped waves occur for a circular hole with a clamped edge, for a hole with a free edge trapped waves occur in all four possible modes of symmetry.