It is shown that localised wave motions (often referred to edge waves or trapped modes) are capable of being supported by two complementary arrangements involving floating ice on water: a finite width ice sheet of constant thickness floating on open water; and an open water channel -- or lead -- embedded in an ice-covered ocean. The search for such solutions is motivated by a simple observation, evidently not made before, that wavelengths of propagating waves in thin ice sheets can be either greater or less than those of the same frequency on an unloaded water surface depending on physical parameters in the problem. The existence of edge waves are confirmed by accurate computations of solutions to integral equations derived from the underlying boundary-value problems using Fourier transform methods.