  We discuss the current status of theoretical and experimental constraints on
the real Higgs singlet extension of the Standard Model. For the second neutral
(non-standard) Higgs boson we consider the full mass range from 1 GeV to 1 TeV
accessible at past and current collider experiments. We separately discuss
three scenarios, namely, the case where the second Higgs boson is lighter than,
approximately equal to, or heavier than the discovered Higgs state at around
125 GeV. We investigate the impact of constraints from perturbative unitarity,
electroweak precision data with a special focus on higher order contributions
to the W boson mass, perturbativity of the couplings as well as vacuum
stability. The latter two are tested up to a scale of 4 x 10^10 GeV using
renormalization group equations. Direct collider constraints from Higgs signal
rate measurements at the LHC and 95% C.L. exclusion limits from Higgs searches
at LEP, Tevatron and LHC are included via the public codes HiggsSignals and
HiggsBounds, respectively. We identify the strongest constraints in the
different regions of parameter space. We comment on the collider phenomenology
of the remaining viable parameter space and the prospects for a future
discovery or exclusion at the LHC.
