  Using Monte Carlo simulations and finite-size scaling, we investigate the XY
antiferromagnet on the triangular, Union Jack and bisected-hexagonal lattices,
and in each case find both Ising and Kosterlitz-Thouless transitions. As is
well-known, on the triangular lattice, as the temperature decreases the system
develops chiral order for temperatures $T < \Tc$, and then quasi-long-range
magnetic order on its sublattices when $T < \Ts$, with $\Ts < \Tc$. The
behavior $\Ts<\Tc$ is predicted by theoretical arguments due to Korshunov,
based on the unbinding of kink-antikink pairs. On the Union Jack and
bisected-hexagonal lattices, by contrast, we find that as $T$ decreases the
magnetizations on some of the sublattices become quasi-long-range ordered at a
temperature $\Ts > \Tc$, before chiral order develops. In some cases, the
sublattice spins then undergo a second transition, of Ising type, separating
two quasi-long-range ordered phases. On the Union Jack lattice, the
magnetization on the degree-4 sublattice remains disordered until $\Tc$ and
then undergoes an Ising transition to a quasi-long-range ordered phase.
