The quickest detection of short-term voltage instability in power systems is considered. The problem is formulated as a Bayesian quickest change-point detection where the pre-change and post-change measurements are non-stationary processes with exponentially decaying and exponentially increasing expectations, respectively. Quickest change detection schemes are proposed and analyzed under both known and unknown post-change models. It is shown that the proposed tests are asymptotically optimal. The results also find applications in instability detection of a general linear system with distinct real eigenvalues.