Morphing aircraft aim to increase the performance of aircraft over multiple flight conditions, by enabling shape changes in flight in order to optimise their aerodynamic properties for the current conditions. The skin of a morphing aircraft is a critical component. It must be compliant in degrees of freedom that are required for actuation, to minimise the actuation loads. However, it must also carry structural loads, and therefore be stiff in load bearing degrees of freedom. This leads to a requirement for extremely anisotropic material systems. A common solution is the use of a corrugated panel. However, previous work on corrugations has not addressed the problem of compressive buckling loads. This work analyses the performance of corrugated panels under buckling loads, and optimises corrugation patterns for the objectives of weight, buckling performance, and actuation compliance. Simplified analytical models that derive properties equivalent to conventional plates are used to obtain approximate estimates of the buckling loads. Furthermore a new mode of buckling, that occurs entirely in-plane and is unique to panels with extreme anisotropy is analysed. The simple models allow optimisation to be performed, and both a single-objective and a multi-objective approach are demonstrated. The results are compared to Finite Element Analysis.