  The seminal theoretical works of Berezinskii, Kosterlitz, and Thouless
presented a new paradigm for phase transitions in condensed matter that are
driven by topological excitations. These transitions have been extensively
studied in the context of two-dimensional XY models -- coupled compasses -- and
have generated interest in the context of quantum simulation. Here, we use a
circuit quantum-electrodynamics architecture to study the critical behavior of
engineered XY models through their dynamical response. In particular, we
examine not only the unfrustrated case but also the fully-frustrated case which
leads to enhanced degeneracy associated with the spin rotational [U$(1)$] and
discrete chiral ($Z_2$) symmetries. The nature of the transition in the
frustrated case has posed a challenge for theoretical studies while direct
experimental probes remain elusive. Here we identify the transition
temperatures for both the unfrustrated and fully-frustrated XY models by
probing a Josephson junction array close to equilibrium using weak microwave
excitations and measuring the temperature dependence of the effective damping
obtained from the complex reflection coefficient. We argue that our probing
technique is primarily sensitive to the dynamics of the U$(1)$ part.
