  We explore simple Higgs-portal models of dark matter (DM) with spin 1/2, 3/2,
and 1, respectively, applying to them constraints from the LUX and PandaX-II
direct detection experiments and from LHC measurements on the 125-GeV Higgs
boson. With only one Higgs doublet, we find that the spin-1/2 DM having a
purely scalar effective coupling to the doublet is viable only in a narrow
range of mass near the Higgs pole, whereas the vector DM is still allowed if
its mass is also close to the Higgs pole or exceeds 1.4 TeV, both in line with
earlier analyses. Moreover, the spin-3/2 DM is in a roughly similar situation
to the spin-1/2 DM, but has surviving parameter space which is even more
restricted. We also consider the two-Higgs-doublet extension of each of the
preceding models, assuming that the expanded Yukawa sector is that of the
two-Higgs-doublet model of type II. We show that in these
two-Higgs-doublet-portal models significant portions of the DM mass regions
excluded in the simplest scenarios by direct search bounds can be reclaimed due
to suppression of the effective DM interactions with nucleons at some ratios of
the $CP$-even Higgs bosons' couplings to the up and down quarks. The regained
parameter space contains areas which can yield a DM-nucleon scattering
cross-section that is far less than its current experimental limit or even goes
below the neutrino-background floor.
