The objective of this project is to investigate prototypical models in shape-memory alloys which display a dichotomy between rigidity and flexibility within the geometrically nonlinear theory of elasticity. We seek to identify critical regularity thresholds in this transition by studying scaling laws in the singular perturbation parameter for models involving an elastic and a surface energy contribution. To this end, we adopt a geometric perspective and explore quantitative rigidity results arising from the multi-well nature of the Cauchy-Green tensors. A particular emphasis will be on the investigation of the highly nonlinear compatibility conditions.