An infinite triangular lattice of classical dipolar spins is usually considered to have a ferromagnetic ground state. We examine the validity of this statement for finite lattices and in the limit of large lattices. We find that the ground state of rectangular arrays is strongly dependent on size and aspect ratio. Three results emerge that are significant for understanding the ground state properties: (i) formation of domain walls is energetically favored for aspect ratios below a critical value; (ii) a vortex state is energetically favored in the thermodynamic limit of an infinite number of spins, but nevertheless such a configuration may not be observed even in very large lattices if the aspect ratio is large; (iii) finite range (R) approximations to actual dipole sums may give spurious results and the limit R ->infinity depends on the way it is taken. For the usual, isotropic limit, the ferromagnetic state is linearly unstable and the domain wall energy is negative for any finite range cutoff.