In the context of instability problems in shells or shell like structures, the objective of the present paper is twofold. Primarily, the paper describes how quasi-static, conservative instability problems can be considered in a multi-parametric context, where generalized path-following procedures for augmented equilibrium problems are used as computational tools. These allow systematic treatment of the higher-dimensional solution sets generated under the variations of certain parameters deemed relevant for the given problem. The efficient implementation of the above mentioned procedures requires however, as an essential ingredient, a non-linear finite element which is not only accurate but also inexpensive. To this end, a systematic view on a corotational Total Lagrangian formulation is described. The TRIC element of Argyris and coworkers is slightly modified, and introduced as core element formulation. Special emphasis is given to the alternative methods for treatment of finite three-dimensional rotations, with reference to both the element definition and solution algorithms. Numerical examples verify the element capabilities, and the possibility to completely describe instability phenomena of large, discretized models.