  We study the lepton portal dark matter (DM) model in which the relic
abundance is determined by the portal coupling among the Majorana fermion DM
candidate $\chi$, the singlet charged scalar mediator $S^\pm$ and the Standard
Model (SM) right-handed lepton. The direct and indirect searches are not
sensitive to this model. This article studies the lepton portal coupling as
well as the scalar portal coupling (between $S^\pm$ and SM Higgs boson), as the
latter is generally allowed in the Lagrangian. The inclusion of scalar portal
coupling not only significantly enhances the LHC reach via the $gg\to h^*\to
S^+S^-$ process, but also provides a few novel signal channels, such as the
exotic decays and coupling deviations of the Higgs boson, offering new
opportunities to probe the model. In addition, we also study the Drell-Yan
production of $S^+S^-$ at future lepton colliders, and find out that the
scenario where one $S^\pm$ is off-shell can be used to measure the lepton
portal coupling directly. In particular, we are interested in the possibility
that the scalar potential triggers a first-order phase transition and hence
provides the stochastic gravitational wave (GW) signals. In this case, the
terrestrial collider experiments and space-based GW detectors serve as
complementary approaches to probe the model.
