Stable problem is a special problem of shell structure, and also a controlling factor in the design of single-layer lattice shell structure (Shen and Chen in Stability of reticulated shell structures, Science Press, Beijing, 1999 [1]), that is, the ultimate bearing capacity of single-layer lattice shell structure is generally determined by the stable bearing capacity. Therefore, revealing the instability mechanism of single-layer lattice shell structure is the basis of stable optimization design, which has important theoretical and engineering value for improving the structural safety reserve and excavating the space spanning ability of single-layer lattice shell structure. In 1991, Blockley et al. (Civ Eng Syst 10:301–317, 1993 [2]; Wu et al. in Civ Eng Syst 10:319–333, 1993 [3]) from Bristol University in the UK proposed the vulnerability theory based on joint well-formedness. Based on the joint well-formedness with clear physical meaning, the structural topological hierarchy model is established based on the clustering process to identify the weakest part of the internal connection of the structure; through the declustering process, the various failure modes with vulnerability of the structure are identified. This method has been widely used in the failure analysis of space frame structure (Agarwal et al. in Struct Saf 23:203–220, 2001 [4]), space rod structure (Zhu and Ye in J Eng Mech 140:112–127, 2013 [5]) and cold-formed steel structure (Ye et al. in Appl Sci 7:182, 2017 [6]). The classical configuration vulnerability theory only focuses on the topological configuration of the structure itself, and cannot consider external factors such as load and constraints; at the same time, the theory analyzes the structural characteristics based on the elastic stiffness matrix under the initial configuration, and cannot consider nonlinearity. However, a large number of research results show that the stability of lattice shell structure is closely related to the characteristics of the structure itself, load mode, support constraints and other conditions, and has strong nonlinearity. In this chapter, the joint well-formedness, an important parameter in the theory of configuration vulnerability, is firstly introduced. Then, on the basis of the classical well-formedness, the geometric stiffness matrix is introduced, and a calculation method of joint well-formedness that can consider load conditions, geometric nonlinearity and support constraints is proposed. By analyzing the changes of joint well-formedness before and after the introduction of the geometric stiffness matrix, the instability mechanism of the single-layer lattice shell structure is revealed from the perspective of the loss of stability. Finally, three single-layer lattice shell structures with different scales are taken as examples to verify the instability mechanism of the single-layer lattice shell structure proposed in this chapter.