  Models with an extended scalar electroweak sector can have vanishing vacuum
expectation values in a basis where an underlying symmetry is imposed. Such
extensions are very well motivated. If a symmetry prevents couplings between
fermions and additional scalars, such scalars could become viable dark matter
candidates if some additional criteria are satisfied. We catalogue
$S_3$-symmetric three-Higgs-doublet models, also allowing for softly broken
$S_3$-symmetric scalar potential terms, based on whether a specific model could
possibly accommodate a dark matter candidate. The variety of the
$S_3$-symmetric family models arises due to different possibilities to arrange
vacuum expectation values. Such models can have vacua with one or two vanishing
vacuum expectation values. In our study we assume that the dark matter
candidate is stabilised by the $\mathbb{Z}_2$ symmetry. The $\mathbb{Z}_2$
symmetry is a remnant of $S_3$ symmetry which survived spontaneous symmetry
breaking, and not superimposed over $S_3$. We explore two models; with an
without CP violation. These models have a single dark and two active scalar
sectors. The active sectors behave in many aspects like a Type-I
two-Higgs-doublet model. The dark matter candidate masses, in two cases, are
different from the known (previously studied) models with three scalar
doublets. After investigating the models in detail, identifying parameters
compatible with both theoretical and experimental constraints, we found that
the dark matter candidate mass could be within the range of
$[52.5,\,89]~\text{GeV}$ or $[6.5,\,44.5]~\text{GeV}$ for a model with CP
violation.
