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In many extensions of the Standard Model (SM) flavour changing neutral current (FCNC) processes can be mediated by tree-level heavy neutral scalars and/or pseudoscalars H-0(A(0)). This generally introduces new sources of flavour violation and CP violation as well as left-handed (LH) and right-handed (RH) scalar (1 -/+ gamma(5)) currents. These new physics (NP) contributions imply a pattern of deviations from SM expectations for FCNC processes that depends only on the couplings of H-0(A(0)) to fermions and on their masses. In situations in which a single H-0 or A(0) dominates NP contributions stringent correlations between Delta F = 2 and Delta F = 1 observables exist. Anticipating the Flavour Precision Era (FPE) ahead of us we illustrate this by searching for allowed oases in the landscape of a given model assuming significantly smaller uncertainties in CKM and hadronic parameters than presently available. To this end we analyze Delta F = 2 observables in B-s,d(0) - (B) over bar (0)(s,d) and K-0 - (K) over bar (0) systems and rare B and K decays with charged leptons in the final state including both left-handed and right-handed scalar couplings of H-0 and A(0) to quarks in various combinations. We identify a number of correlations between various flavour observables that could test and distinguish these different scenarios. The prominent role of the decays B-s,B-d -> mu(+)mu(-) in these studies is emphasized. Imposing the existing flavour constraints, a rich pattern of deviations from the SM expectations in rare B-s,B-d decays emerges provided M-H <= 1 TeV. NP effects in rare K decays, except for K-L -> mu(+)mu(-) , turn out to be very small. In K-L -> mu(+)mu(-) they can be as large as the SM contributions but due to hadronic uncertainties this is still insufficient to learn much about new scalars from this decay in the context of models considered here. Flavour violating SM Higgs contributions to rare B-d and K decays turn out to be negligible once the constraints from Delta F = 2 processes are taken into account. But B (B -> mu(+)mu(-) ) can still be enhanced up to 8%. Finally, we point out striking differences between the correlations found here and in scenarios in which tree-level FCNCs are mediated by a new neutral gauge boson Z’.