This work studies the solid-shell finite element approach to approximate thin structures using a stabilised mixed displacement-stress formulation based on the Variational Multiscale framework. First, the numerical locking effects inherent to the solid-shell approach are characterised using a variety of benchmark problems in the infinitesimal strain approximation. Later, the results are extended to formulate the mixed approach in finite strain hyperelastic problems. In the first part, the stabilised mixed displacement-stress formulation is proven to be adequate to deal with all kinds of numerical locking. Additionally, a more comprehensive analysis of each individual type of numerical locking, how it is triggered and how it is overcome is also provided. The numerical locking usually occurs when parasitic strains overtake the system of equations through specific components of the stress tensor. To properly analyse them, the direction of each component of the stress tensor has been defined respect to the shell directors. Therefore, it becomes necessary to formulate the solid-shell problem in curvilinear coordinates, allowing to give mechanical meaning to the stress components (shear, twisting, membrane and thickness stresses) independently of the global frame of reference. The conditions in which numerical locking is triggered as well as the stress tensor component responsible of correcting the locking behaviour have been identified individually by characterising the numerical response of a set of different benchmark problems. In the second part, the formulation is extended to finite strain solid dynamics involving hyperelastic materials. The aim of introducing this method is to obtain a robust stabilised mixed formulation that enhances the accuracy of the stress field. This improved formulation holds great potential for accurately approximating shell structures undergoing finite deformations. The accuracy of the stress field is successfully enhanced while maintaining the accuracy of the displacement field. These improvements are also inherited to the solid-shell elements, providing locking-free approximation of thin structures.