This contribution presents a model order reduction framework for real-time efficient solution of trimmed, multi-patch isogeometric Kirchhoff-Love shells. In several scenarios, such as design and shape optimization, multiple simulations need to be performed for a given set of physical or geometrical parameters. This step can be computationally expensive in particular for real world, practical applications. We are interested in geometrical parameters and take advantage of the flexibility of splines in representing complex geometries. In this case, the operators are geometry-dependent and generally depend on the parameters in a non-affine way. Moreover, the solutions obtained from trimmed domains may vary highly with respect to different values of the parameters. Therefore, we employ a local reduced basis method based on clustering techniques and the Discrete Empirical Interpolation Method to construct affine approximations and efficient reduced order models. In addition, we discuss the application of the reduction strategy to parametric shape optimization. Finally, we demonstrate the performance of the proposed framework to parameterized Kirchhoff-Love shells through benchmark tests on trimmed, multi-patch meshes including a complex geometry. The proposed approach is accurate and achieves a significant reduction of the online computational cost in comparison to the standard reduced basis method.