The solid-liquid phase diagram of two-dimensional, laterally confined systems made of interacting parallel dipoles normal to the plane is investigated by Monte Carlo method. The lowest-energy solid phase is relaxed from a regular triangular lattice. This Wigner crystal phase keeps the main features of the original monocrystal. On the other hand, the solid phase relaxed from a random initial configuration has a slightly higher energy and is called Wigner glass. It exhibits randomly oriented crystallites separated by chains of topological defects, namely alternate 5- and 7-nearest-neighbour sites. These chains start to move within the solid phase at a temperature lower than the bulk melting temperature.