  The Higgs sector of the Next-to-Minimal Two-Higgs-Doublet Model (N2HDM) is
obtained from the Two-Higgs-Doublet Model (2HDM) containing two complex Higgs
doublets, by adding a real singlet field. In this paper, we analyse the vacuum
structure of the N2HDM with respect to the possibility of vacuum instabilities.
We show that while one type of charge- and CP-preserving vacuum cannot coexist
with deeper charge- or CP-breaking minima, there is another type of vacuum
whose stability is endangered by the possible occurrence of deeper charge- and
CP-breaking minima. Analytical expressions relating the depth of different
vacua are deduced. Parameter scans of the model are carried out that illustrate
the regions of parameter space here the vacuum is either stable or metastable
as well as the regions where tunnelling to deeper vacua gives rise to a too
short lifetime of the vacuum. Taking other experimental and theoretical
constraints into account, we find that the vacuum stability constraints have an
important impact on the phenomenology of the N2HDM.
