  We analyse an array of linearly extended monodomain dipoles forming square
and kagome lattices. We find that its phase diagram contains two (distinct)
finite-entropy kagome ice regimes - one disordered, one algebraic - as well as
a low-temperature ordered phase. In the limit of the islands almost touching,
we find a staircase of corresponding entropy plateaux, which is analytically
captured by a theory based on magnetic charges. For the case of a modified
square ice array, we show that the charges ('monopoles') are excitations
experiencing two distinct Coulomb interactions: a magnetic 'three-dimensional'
one as well as a logarithmic `two dimensional' one of entropic origin.
