  We simulate a trapped quasi-two-dimensional Bose gas using a classical field
method. To interpret our results we identify the uniform
Berezinskii-Kosterlitz-Thouless (BKT) temperature $T_{BKT}$ as where the system
phase space density satisfies a critical value. We observe that density
fluctuations are suppressed in the system well above $T_{BKT}$ when a
quasi-condensate forms as the first occurrence of degeneracy. At lower
temperatures, but still above $T_{BKT}$, we observe the development of
appreciable coherence as a prominent finite-size effect, which manifests as
bimodality in the momentum distribution of the system.
  At $T_{BKT}$ algebraic decay of off-diagonal correlations occurs near the
trap center with an exponent of 0.25, as expected for the uniform system.
  Our results characterize the low temperature phase diagram for a trapped
quasi-2D Bose gas and are consistent with observations made in recent
experiments.
