Thin elastic sheets are usually modeled by variational problems for an energy with two scales, a (strong) stretching energy and a (weak) bending energy. A useful paradigm in understanding the behavior of theses sheets under various loadings/boundary conditions is the following -- Singularites/microstructure in the observed configurations of thin sheets reflect "geometric incompatibility", that is the non-existence of admissible, sufficiently smooth isometric immersions (zero stretching energy test functions). Using specific examples, and a combination of rigorous results and conjectures motivated by numerical computation, I will try to argue that the relation between the observed geometry of thin elastic sheets and the existence/nonexistence of isometric immersions is much more subtle.