  We explore three-Higgs-doublet models that may accommodate scalar Dark Matter
where the stability is based on an unbroken $U(1)$-based symmetry, rather than
the familiar $\mathbb{Z}_2$ symmetry. Our aim is to classify all possible ways
of embedding a $U(1)$ symmetry in a three-Higgs-doublet model. The different
possibilities are presented and compared. All these models contain
mass-degenerate pairs of Dark Matter candidates due to a $U(1)$ symmetry
unbroken (conserved) by the vacuum. Most of these models preserve CP. In the
CP-conserving case the pairs can be seen as one being even and the other being
odd under CP or as having opposite charges under $U(1)$. Not all symmetries
presented here were identified before in the literature, which points to the
fact that there are still many open questions in three-Higgs-doublet models. We
also perform a numerical exploration of the $U(1) \otimes U(1)$-symmetric 3HDM,
this is the most general phase-invariant (real) three-Higgs-doublet model. The
model contains a multi-component Dark Matter sector, with two independent mass
scales. After imposing relevant experimental constraints we find that there are
possible solutions throughout a broad Dark Matter mass range, 45-2000 GeV, the
latter being a scan cutoff.
