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It is well known that buckling instabilities occur when elastic solids are subject to compressive stresses. However, this does not preclude the occurrence of instabilities in systems subject to global tensile loads. Such tensile instabilities may be caused by certain discontinuities (geometrical or material) which re-distribute the stresses applied on its boundary, generating local compressive stresses inside the solid. This research deals with the tensile instabilities in elastic solids by using linear bifurcation analysis, which leads to eigenvalue problems. Then the links between mechanics and mathematics in these tensile instabilities/eigen-systems are demonstrated by using a combination of both numerical and asymptotic analyses. Three main problems have been investigated: a hybrid energy method on edge-buckling, the tensile wrinkling of a stretched bi-annular plate, and the tensile instabilities developed in a radially stretched thick cylindrical tube. We start by recording a coordinate-free derivation for Föppl–von Kármán equations and the corresponding bifurcation equations (both weak forms and strong forms with their boundary conditions) based on minimum energy principle. This set of equations is applicable to anisotropic elastic thin plates in any planar geometries, which is then specialised for the bifurcation problem of isotropic elastic plates, and a further case under in-plane loading. In the first main problem, we propose a hybrid energy method which provides an accurate and computationally efficient algorithm for the instability analysis of a class of edge-buckling problems. This algorithm is based on the existing simplest asymptotic approximations and an energy principle (weak form for bifurcation). Fairly accurate and robust approximations can be achieved for both the critical buckling load and mode number even though the simplest asymptotic ansatz is employed. We also explore a number of additional mathematical features that have an intrinsic interest in the context of multi-parameter eigenvalue problems. Then we consider the wrinkling instabilities of a stretched bi-annular plate, which consists of two fully bonded concentric annuli with different mechanical properties. The effects of the mechanical and geometrical parameters on critical wrinkling are studied, using both numerical and asymptotic techniques. It is found that the critical external buckling loading, the wrinkle numbers and the wrinkled-shape can behave completely differently compared with the singleannular case. The influence of discontinuities (the interface between these two annuli) on localised instabilities is also illustrated thoroughly. Finally, a WKB analysis is conducted