We perform a combined and extended analysis of effective scalar, vector, Majorana and Dirac fermion Higgs portal models of dark matter (DM). In these models, DM couples to the Standard Model (SM) Higgs boson via an operator of the form $\mathcal{O}_{\mathrm{DM}}\, H^\dagger H$. For the fermion models, we allow an admixture of the scalar $\overline{\psi} \psi$ and pseudoscalar $\overline{\psi} i\gamma_5 \psi$ interaction terms. On each model, we impose constraints from the Planck measured DM relic density, LHC limits on the Higgs invisible branching ratio, indirect search limits from the WMAP7/Planck observations of the cosmic microwave background, a combined analysis of 15 dwarf spheroidal galaxies by Fermi-LAT and the upcoming Cherenkov Telescope Array (CTA) as well as the direct search limits from the LUX and projected XENON1T experiments. We find in line with previous studies that the direct search experiments will continue to exclude large portions of the model parameter space. Due to the momentum suppression of the direct search cross section in the case of a pure pseudoscalar interaction of a fermionic DM candidate, future indirect searches are the only class of experiments capable of probing the high mass range of the theory.