This article gives a concise overview of the development and current status of studies on models of the early Universe without an initial singularity, namely the cosmological bounce and genesis scenarios, constructed within a broad class of scalar-tensor theories, specifically Horndeski theories and their generalizations. The review focuses on topics related to linear stability at the perturbation level against the background of nonsingular cosmological solutions: (1) the no-go theorem, valid for nonsingular cosmologies within the Hornde„ski theory, (2) updates on possible approaches to evade the no-go theorem, (3) the role of disformal transformations relating the Horndeski subclasses in generalized theories like DHOST, (4) effects on stability caused by additional matter coupling and the potential emergence of superluminal perturbation modes in the multi-component setting.