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The simplest extension of the Standard Model (SM) that generally introduces new sources of flavour violation and CP violation as well as right-handed (RH) currents is the addition of a U(1) gauge symmetry to the SM gauge group. If the corresponding heavy gauge boson (Z’) mediates FCNC processes in the quark sector at tree-level, these new physics (NP) contributions imply a pattern of deviations from SM expectations for FCNC processes that depends only on the couplings of Z’ to fermions and on its mass. This implies stringent correlations between Delta F = 2 and Delta F = 1 observables which govern the landscape of the allowed parameter space for Z’-models. Anticipating the Flavour Precision Era (FPE) ahead of us we illustrate this by searching for allowed oases in this landscape assuming significantly smaller uncertainties in CKM and hadronic parameters than presently available. To this end we analyze Delta F = 2 observables in K-0 - (K) over bar (0) and B-s,d(0) - (B) over bar (0)(s,d) systems and rare K and B decays including both left-handed and right-handed Z’-couplings to quarks in various combinations. We identify a number of correlations between various flavour observables that could test and distinguish these different Z’ scenarios. The important role of b -> sl(+)l(-) and b -> s nu(nu) over bar transitions in these studies is emphasized. Imposing the existing flavour constraints, a rich pattern of deviations from the SM expectations in B-s,B-d and K meson systems emerges provided M-Z’ <= 3 TeV. While for M-Z’ >= 5 TeV Z’ effects in rare B-s,B-d decays are found typically below 10% and hard to measure even in the FPE, K -> pi nu(nu) over bar and K-L -> pi(0)l(+)l(-) decays provide an important portal to scales beyond those explored by the LHC. We apply our formalism to NP scenarios with induced flavour changing neutral Z-couplings to quarks. We find that in the case of B-d and K decays such Z-couplings still allow for sizable departures from the SM. On the other hand in the B-s system, constraints on b -> sl(+)l(-) transitions basically eliminate NP effects from such couplings.