Flexible displays present a challenging problem in terms of mechanical integrity, a result of the considerable hygro-thermomechanical contrast between the inorganic, brittle device layers and the compliant polymer substrates. This paper reviews the main approaches to study and identify the key factors, which control the mechanical stability of this class of displays. Focus is put on the analyses of residual stress and damage under tensile loading. Novel electro-mechanical methods are used for accurate insight into critical phenomena. An important result is that the thickness and stiffness of the substrate control the critical strain for failure of the device layers.