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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 number of IDM scenarios, which are consistent with current constraints on direct detection 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 A~H $ and $e^+e^- \to H^+H^-$, followed by decays of $A$ and $H^\pm$ into final states which always 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 from 250 GeV up to 3 TeV. For low mass scenarios, high significance can be obtained for the signal signatures with two muons or an electron and a muon in the final state. For high mass scenarios, which are only accessible at high energy stages of CLIC, the significance is too low for the leptonic signature and the semi-leptonic final state has to be used as the discovery channel. Results presented for this channel are based on the fast simulation of the CLIC detector response with the DELPHES package.