  We study the triangular-lattice Ising model with dipolar interactions,
inspired by its realisation in artificial arrays of nanomagnets. We show that a
classical spin-liquid forms at intermediate temperatures, and that its
behaviour can be tuned by temperature and/or a small lattice distortion between
a string Luttinger liquid and a domain-wall-network state. At low temperature
there is a transition into a magnetically ordered phase, which can be
first-order or continous with a crossover in the critical behaviour between
Pokrovsky-Talapov and 2D-Ising universality. When the Pokrovsky-Talapov
criticality dominates, the transition is essentially of the Kasteleyn type.
