Beyond the Standard Model (SM) extensions usually include extended Higgs sectors. Models with singlet or doublet fields are the simplest ones that are compatible with the ρ parameter constraint. The discovery of new non-SM Higgs bosons and the identification of the underlying model requires dedicated Higgs properties analyses. In this paper, we compare several Higgs sectors featuring 3 CP-even neutral Higgs bosons that are also motivated by their simplicity and their ability to solve some of the flaws of the SM. They are: the SM extended by a complex singlet field (CxSM), the singlet extension of the 2-Higgs-Doublet Model (N2HDM), and the Next-to-Minimal Supersymmetric SM extension (NMSSM). In addition, we analyse the CP-violating 2-Higgs-Doublet Model (C2HDM), which provides 3 neutral Higgs bosons with a pseudoscalar admixture. This allows us to compare the effects of singlet and pseudoscalar admixtures. Through dedicated scans of the allowed parameter space of the models, we analyse the phenomenologically viable scenarios from the view point of the SM-like Higgs boson and of the signal rates of the non-SM-like Higgs bosons to be found. ... mehrIn particular, we analyse the effect of singlet/pseudoscalar admixture, and the potential to differentiate these models in the near future. This is supported by a study of couplings sums of the Higgs bosons to massive gauge bosons and to fermions, where we identify features that allow us to distinguish the models, in particular when only part of the Higgs spectrum is discovered. Our results can be taken as guidelines for future LHC data analyses, by the ATLAS and CMS experiments, to identify specific benchmark points aimed at revealing the underlying model.