  We study the finite-temperature behaviour of two-dimensional S=1/2 Heisenberg
antiferromagnets with very weak easy-axis and easy-plane exchange anisotropies.
By means of quantum Monte Carlo simulations, based on the continuous-time loop
and worm algorithm, we obtain a rich set of data that allows us to draw
conclusions about both the existence and the type of finite-temperature
transition expected in the considered models. We observe that the essential
features of the Ising universality class, as well as those of the
Berezinskii-Kosterlitz-Thouless (BKT) one, are preserved even for anisotropies
as small as 10^{-3} times the exchange integral; such outcome, being referred
to the most quantum case S=1/2, rules out the possibility for quantum
fluctuations to destroy the long or quasi-long range order, whose onset is
responsible for the Ising or BKT transition, no matter how small the
anisotropy. Besides this general issue, we use our results to extract, out of
the isotropic component, the features which are peculiar to weakly anisotropic
models, with particular attention for the temperature region immediately above
the transition. By this analysis we aim to give a handy tool for understanding
the experimental data relative to those real compounds whose anisotropies are
too weak for a qualitative description to accomplish the goal of singling out
the genuinely two-dimensional critical behaviour.
