Blistering occurs when a thin solid layer locally separates from an underlying substrate through cracking of a bulk material, delamination of a composite material, or peeling of a membrane adhered to the substrate by a thin layer of viscous fluid. In this last scenario, the expansion of the newly formed blister by fluid injection occurs via a displacement flow, which peels apart the adhered surfaces through a two-way interaction between flow and deformation. Such blisters are prone to fluid and solid mechanical instabilities. If the injected fluid is less viscous than the fluid already occupying the gap, patterns of short and stubby fingers form on the propagating fluid interface. This process is regulated by membrane compliance, which if increased delays the onset of fingering to higher flow rates and reduces finger amplitude. Suppression is mediated by the locally tapered geometry of the blister near the fluid interface, which is imposed by the underlying blistering flow. Buckling/wrinkling instabilities of the delaminated layer arise for sufficiently thin membranes and can interact with the fluid mechanical fingering instability.