Classification of Dark Matter (DM) models is the key for the consistent exploration of DM model space and comprehensive probe of DM nature at current and future experiments. We perform systematic classification of Minimal Consistent DM (MCDM) models -- gauge invariant, renormalisable and anomaly free models -- in terms of DM and mediator weak multiplets. MCDM represents a natural building block for any BSM theory which includes DM. We consider two cases: 1) when only SM fields play the role of the mediator for DM multiplet; 2) when there is an additional mediator multiplet -- odd or even under the symmetry, providing DM stability. We explore the phenomenology of new representative model found in this classification. This model has a two-component dark-sector containing an odd singlet Dirac fermion alongside an accidentally stable even pseudo-scalar mediator. The model exhibits a non-trivial interplay between two components of DM contributing to relic density and can be probed at direct, indirect DM search experiments as well as at the LHC and future FCC-hh colliders.