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Pattern formation in reaction-diffusion systems is of great importance in surface micropatterning [Grzybowski et al., Soft Matter 1, 114 (2005)], self-organization of cellular micro-organisms [Schulz et al., Annu. Rev. Microbiol. 55, 105 (2001)], and in developmental biology [Barkai et al., FEBS Journal 276, 1196 (2009)]. In this work, we apply the lattice Boltzmann method to study pattern formation in reaction-diffusion systems. As a first methodological step, we consider the case of a single species undergoing transformation reaction and diffusion. In this case, we perform a third-order Chapman-Enskog multiscale expansion and study the dependence of the lattice Boltzmann truncation error on the diffusion coefficient and the reaction rate. These findings are in good agreement with numerical simulations. Furthermore, taking the Gray-Scott model as a prominent example, we provide evidence for the maturity of the lattice Boltzmann method in studying pattern formation in nonlinear reaction-diffusion systems. For this purpose, we perform linear stability analysis of the Gray-Scott model and determine the relevant parameter range for pattern formation. Lattice Boltzmann simulations allow us not only to test the validity of the linear stability phase diagram including Turing and Hopf instabilities, but also permit going beyond the linear stability regime, where large perturbations give rise to interesting dynamical behavior such as the so-called self-replicating spots. We also show that the length scale of the patterns may be tuned by rescaling all relevant diffusion coefficients in the system with the same factor while leaving all the reaction constants unchanged.