Determination of the nature of Dark Matter (DM) is one of the most fundamental problems of particle physics and cosmology. If DM is light enough and interacts with Standard Model particles directly or via some mediators with a strength beyond the gravitational one, it can be directly produced at the Large Hadron Collider or future particle accelerators. The typical signature from DM produced in particles collisions is missing transverse energy, MET, due to the fact that they escape undetected from the experimental apparatus. We present study for the complete set of dimension 5 and dimension 6 effective operators involving scalar, fermion and vector DM to explore the possibility to distinguish these operators and characterise the spin of DM. We have found that, depending on the spin of the DM, the DM parts of the effective operators lead to a different energy dependence of the cross-sections and to different distributions of the invariant mass of the DM pair, and consequently to different MET distributions. We have found that at the LHC with high luminosity, certain classes of EFT operators can be distinguished from each other and, through this, it is possible to characterise the spin of DM in some cases. This study can be directly applied beyond the EFT paradigm.