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The inflation of hyperelastic thin shells is an important and highly nonlinear problem that arises in multiple engineering applications involving severe kinematic and constitutive nonlinearities in addition to various instabilities. We present an isogeometric approach to compute the inflation of hyperelastic thin shells, following the Kirchhoff-Love hypothesis and associated large deformation. Both the geometry and the deformation field are discretized using Catmull-Clark subdivision bases which provide the C1-continuous finite element framework required for the Kirchhoff-Love shell formulation. To follow the complex nonlinear response of hyperelastic thin shells, the inflation is simulated incrementally, and each incremental step is solved via the Newton-Raphson method enriched with arc-length control. Eigenvalue analysis of the linear system after each incremental step allows for inducing bifurcation to a lower energy mode in case stability of the equilibrium is lost. The proposed method is first validated using benchmarks, and then applied to engineering applications, where we demonstrate the ability to simulate large deformation and associated complex instabilities.