Motivated by applications such as gecko-inspired adhesives and microdevices featuring slender rodlike bodies, there has been an increase in interest in the deformed shapes of elastic rods adhering to rigid surfaces. A central issue in analyses of the rod-based models for these systems is the stability of the predicted equilibrium configurations. In this talk, we discuss the recent development of nonlinear criteria to access stability. Surprisingly, these criteria arose from work examining the buckling of tree-like structures and the human spine. The criteria feature Riccati equations and exploit a classic construction by Legendre in his examination of the second variation of a functional. The talk is based on joint work with my former students Tim Tresierras and Daniel Peters along with Jeffrey Lotz (UCSF), Carmel Majidi (CMU), and John Williams (Cambridge University).