  We consider the issue of vacuum stability and triviality bound of the singlet
extension of the Standard Model (SM) with a singlet fermion dark matter (DM).
In this model, the singlet scalar plays the role of a messenger between the SM
sector and the dark matter sector. This model has two Higgs-like scalar bosons,
and is consistent with all the data on electroweak precision tests, thermal
relic density of DM and its direct detection constraints. We show that this
model is stable without hitting Landau pole up to Planck scale for 125 GeV
Higgs boson. We also perform a comprehensive study of vacuum structure, and
point out that a region where electroweak vacuum is the global minimum is
highly limited. In this model, both Higgs-like scalar bosons have reduced
couplings to the SM weak gauge bosons and the SM fermions, because of the
mixing between the SM Higgs boson and the singlet scalar. There is also a
possibility of their invisible decay(s) into a pair of DM's. Therefore this
model would be disfavored if the future data on the $(\sigma \cdot B)_{VV}$ or
$(\sigma \cdot B)_{f\bar{f}}$ with $V=\gamma,W,Z$ and $f=b, \tau$ turn out
larger than the SM predictions.
