The well-known Saffman-Taylor instability, characterized by the formation of finger-like structures when a less viscous fluid is injected into a more viscous one within a narrow gap between two solid surfaces, is extended to shear-thinning fluid flows. Specifically, we examine the displacement of a viscous Newtonian oil by a shear-thinning fluid in a radial geometry. I will present the results of some preliminary experiments that show how these fingers grow during the early stages and are subsequently suppressed at later stages. The conditions for the onset of instability will be derived theoretically. We will then consider another type of instability which arises when a shear-thinning fluid is injected into a lubricated gap, resulting in the formation of fractures, some of which evolve to form branch-like structures. The origin of this second instability is of an entirely different nature and occurs in regimes where shear stress is negligible in comparison to extensional stress. I will contrast this instability with Saffman-Taylor instabilities and conclude with an experiment that illustrates that both instabilities can develop simultaneously.