Ultimately, the eventual extinction of any biological population is an inevitable outcome. While extensive research has focused on the average time it takes for a population to go extinct under various circumstances, there has been limited exploration of the distributions of extinction times and the likelihood of significant fluctuations. Recently, Hathcock and Strogatz [D. Hathcock and S. H. Strogatz, Phys. Rev. Lett. 128, 218301 (2022)0031-900710.1103/PhysRevLett.128.218301] identified Gumbel statistics as a universal asymptotic distribution for extinction-prone dynamics in a stable environment. In this study we aim to provide a comprehensive survey of this problem by examining a range of plausible scenarios, including extinction-prone, marginal (neutral), and stable dynamics. We consider the influence of demographic stochasticity, which arises from the inherent randomness of the birth-death process, as well as cases where stochasticity originates from the more pronounced effect of random environmental variations. Our work proposes several generic criteria that can be used for the classification of experimental and empirical systems, thereby enhancing our ability to discern the mechanisms governing extinction dynamics. Employing these criteria can help clarify the underlying mechanisms driving extinction processes.