Thin structures typically buckle out of plane when compressed, giving rise to a myriad of shapes. These range from classic Euler buckling to honey coiling to microscale wrinkling patterns, and in some cases correspond to failure, but in others may be exploited for applications such as flexible electronics. In this talk I will present two examples of compression-driven out-of-plane deformation in thin sheets. The first is a failure mode in the manufacture of glass sheets, and the second concerns the dynamic wrinkling of thin elastic sheets under confinement. This talk will involve a combination of experimental results and mathematical modelling.