In this note, an extension of Khasminskii’s theorem on almost sure stability of linear stochastic differential equations to a class of nonlinear stochastic deferential equations is presented. The necessary and sufficient conditions for almost sure stability (a.s.) are proved. It is shown that in the second-order case, the stable region can be exactly determined by studying the singular boundaries of one-dimensional diffusion processes. The authors present a modified form of Feller’s criteria for classification of singular boundaries (c.s.b.). The new criteria are equivalent to and much simpler for applications than Feller’s criteria. Two examples of nonlinear stochastic dynamic systems with stable regions illustrate the application procedures.