By applying differential form theory, we consider the singular control problem for non-linear systems with control variables appearing linearly in both the system dynamics and the performance index. First, we derive necessary conditions of singular optimality for a single-input system, including the relation to the Euler-Poisson equation and to the generalized Legendre-Clebsch condition. Defining thedegree of singularity,we develop necessary conditions satisfied by the singular trajectory embedded in a reduced space. For a time-invariant system, we clarify the relation between the dynamic and the related static optimality. Second, we derive necessary conditions for singular optimality for a multi-input system where the dimension of the control vector is equal to that of the state space. We show that the Shima-Sawaragi condition for the optimality of boundary controls and the generalized Legendre-Clebsch condition are obtained from these conditions. The results are also applied to the analysis of a time-invariant system.