  The Inert Doublet Model (IDM) is one of the simplest extensions of the
Standard Model (SM), providing a dark matter candidate. It is a two Higgs
doublet model with a discrete $Z_2$ symmetry, that prevents the scalars of the
second doublet (inert scalars) from coupling to the SM fermions and makes the
lightest of them stable. We study a large group of IDM scenarios, which are
consistent with current constraints on direct detection, including the most
recent bounds from the XENON1T experiment and relic density of dark matter, as
well as with all collider and low-energy limits. We propose a set of benchmark
points with different kinematic features, that promise detectable signals at
future $e^+e^-$ colliders. Two inert scalar pair-production processes are
considered, $e^+e^- \to H^+H^-$ and $e^+e^- \to AH$, followed by decays of
$H^\pm$ and $A$ into final states which include the lightest and stable neutral
scalar dark matter candidate $H$. Significance of the expected observations is
studied for different benchmark models and different running scenarios, for
centre-of-mass energies up to 3 TeV. Numerical results are presented for the
signal signatures with two muons or an electron and a muon in the final state.
For high mass scenarios, when the significance is too low for the leptonic
signatures, the semi-leptonic signature can be used as the discovery channel.
