The Gray-Scott model is a reaction-diffusion system that is known to exhibit complicated spatial patterns. These include: stripes, rings, spots, domain-filling curves and any combination thereof. In this talk, we consider the stripe solutions in two dimensions. Such a solution can exhibit three different types of instability: a splitting instability, whereby a stripe self-replicates into two parallel stripes; a breakup instability, where a stripe breaks up into spots; and a zigzag instability, whereby a stripe develops a wavy perturbation in the transversal direction. We derive explicit thresholds for all three types of instability. Some open problems will be discussed.