We present an overview of selected copper-based quasi-2D square-lattice spin-1/2 materials with an easy-plane anisotropy, providing the possibility to study emergent Berezinskii-Kosterlitz-Thouless (BKT) correlations. In particular, in those materials with a comparatively small exchange coupling, the effective XY anisotropy of the low-temperature spin correlations can be controlled by an applied magnetic field, yielding a systematic evolution of the BKT correlations. In cases where the residual interlayer correlations are small enough, dynamical BKT correlations in the critical regime may be observed experimentally, whereas the completion of the genuine BKT transition is preempted by the onset of long-range order.