The N2HDM is based on the CP-conserving 2HDM extended by a real scalar singlet field. Its enlarged parameter space and its fewer symmetry conditions as compared to supersymmetric models allow for an interesting phenomenology compatible with current experimental constraints, while adding to the 2HDM sector the possibility of Higgs-to-Higgs decays with three different Higgs bosons. In this paper the N2HDM is subjected to detailed scrutiny. Regarding the theoretical constraints we implement tests of tree-level perturbativity and vacuum stability. Moreover, we present, for the first time, a thorough analysis of the global minimum of the N2HDM. The model and the theoretical constraints have been implemented in ScannerS, and we provide N2HDECAY, a code based on HDECAY, for the computation of the N2HDM branching ratios and total widths including the state-of-the-art higher order QCD corrections and off-shell decays. We then perform an extensive parameter scan in the N2HDM parameter space, with all theoretical and experimental constraints applied, and analyse its allowed regions. We find that large singlet admixtures are still compatible with the Higgs data and investigate which observables will allow to restrict the singlet nature most effectively in the next runs of the LHC. ... mehrSimilarly to the 2HDM, the N2HDM exhibits a wrong-sign parameter regime, which will be constrained by future Higgs precision measurements.