In this paper, we study modulation instabilities (MI) in a one-dimensional chain configuration of a flexible mechanical metamaterial (flexMM). Using the lumped element approach, flexMMs can be modeled by a coupled system of discrete equations for the longitudinal displacements and rotations of the rigid mass units. In the long wavelength regime, and applying the multiple-scales method we derive an effective nonlinear Schr\"odinger equation for slowly varying envelope rotational waves. We are then able to establish a map of the occurrence of MI to the parameters of the metamaterials and the wavenumbers. We also highlight the key role of the rotation-displacement coupling between the two degrees of freedom in the manifestation of MI. All analytical findings are confirmed by numerical simulations of the full discrete and nonlinear lump problem. These results provide interesting design guidelines for nonlinear metamaterials offering either stability to high amplitude waves, or conversely being good candidates to observe instabilities.