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The coming flavour precision era will allow to uncover various patterns of flavour violation in different New Physics (NP) scenarios that are presently washed out by hadronic and experimental uncertainties. We illustrate this by performing the anatomy of flavour violation in the 331 models, based on the gauge group SU(3)(C) x SU(3)(L) x U(1)(X) that are among the simplest NP scenarios with new sources of flavour and CP violation. The latter originate dominantly through the flavour violating interactions of ordinary quarks and leptons with a new heavy Z’ gauge boson. After discussing first these models in some generality, we present a detailed study of Delta F = 2 observables and of rare K and B meson decays in the specific "beta = 1/root 3" model (to be called (331) over bar model) assuming significantly smaller uncertainties in CKM and hadronic parameters than presently available. The most prominent roles in our analysis play epsilon(K), Delta M-q (q = d, s), the mixing induced CP asymmetries S-psi Ks and S-psi phi, rare decays B-s,B-d -> mu(+)mu(-) and in particular the CP-asymmetry S-mu+mu-(s) in B-s -> mu(+)mu(-). As the phenomenology of Z’ contributions is governed only by M-Z’ and four new parameters, (S) over tilde (13), (S) over tilde (23), delta(1) and delta(2), we identify a number of correlations between various observables that differ from those known from CMFV models. While, the Delta F = 2 observables allow still for four oases in the new parameter space, we demonstrate how the inclusion of Delta F = 1 observables can in the future identify the optimal oasis for this model. Favouring the inclusive value of vertical bar V-ub vertical bar, for 1 TeV <= M-Z’ <= 3 TeV, the (331) over bar model is in good agreement with the recent data for B+ -> tau(+)nu(tau), it removes the epsilon(K) - S-psi KS tension present in the SM and the epsilon(K) - Delta M-s,M-d tension present in CMFV models. Simultaneously, while differing from the SM, it is consistent with the present data on B(B-s,B-d -> mu(+)mu(-)) and S-psi phi. We identify the triple correlation B(B-s -> mu(+)mu(-)) - S-psi phi - S-mu+mu-(s) that constitutes an important test of this NP scenario.