  We study the phenomenology of the Standard Model (SM) Higgs sector extended
by two singlet scalars. The model predicts two CP-even scalars $h_{1,2}$ which
are a mixture of doublet and singlet components as well as a pure singlet
scalar $S_{0}$ which is a dark matter candidate. We show that the model can
satisfy the relic density and direct detection constraints as well as all the
recent ATLAS and CMS measurements. We also discuss the effect of the extra
Higgs bosons on the different Higgs triple couplings $h_{i}h_{j}h_{k}$,
$i,j,k=1,2$. A particular attention is given to the triple self-coupling of the
SM-like Higgs where we found that the one loop corrections can reach $150\%$ is
some cases. We also discuss some production mechanisms for $h_{1}$ and $h_{2}$
at the LHC as well as at the future International Linear Collider. It is found
that the production cross section of a pair of SM-like Higgs bosons could be
much larger than the corresponding one in the SM and would reveal physics
beyond the SM if observable. We also show that in this model the branching
ratio of the SM-like Higgs decaying to two singlet scalars could be of the
order of $20\%$, therefore the production of the SM Higgs followed by its decay
to a pair of singlets would be an important source of production of singlet
scalars.
