In this talk, I will present my joint-work (with Haisen Zhang) on second-order necessary conditions for the stochastic optimal control problem, in the case that the control variable enters into both the drift and the diffusion terms. When the control region is convex, a pointwise second-order necessary condition for stochastic singular optimal controls in the classical sense is established; while for the general case, we derive a pointwise second-order necessary condition for stochastic singular optimal controls in the sense of Pontryagin-type maximum principle. Unlike the classical first-order necessary optimality condition, it is found that the corrected term of the second-order adjoint equation plays a crucial role in the formulation of the second-order necessary optimality conditions for both the convex and non-convex setting.