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This work considers a two-dimensional hyperbolic reaction–diffusion system with different inertia and explores criteria for various instabilities, like a wave, Turing, and Hopf, both theoretically and numerically. It is proven that wave instability may occur in a two-species hyperbolic reaction–diffusion system with identical inertia if the diffusion coefficients of the species are nonidentical but cannot occur if diffusion coefficients are identical. Wave instability may also arise in a two-dimensional hyperbolic reaction–diffusion system if the diffusivities of the species are equal, which is never possible in a parabolic reaction–diffusion system, provided the inertias are different. Interestingly, Turing instability is independent of inertia, but the stability of the corresponding local system depends on the inertia. Theoretical results are demonstrated with an example where the local interaction is represented by the Schnakenberg system.