Status of the Higgs Singlet Extension of the Standard Model after LHC Run 1

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.