  We give a Kodaira-type classification of general singular fibers of a
holomorphic Lagrangian fibration in Fujiki's class $\mathcal C$. Our approach
is based on the study of the characteristic vector field of the discriminantal
hypersurface, which naturally arises from the defining equation of the
hypersurface via the symplectic form. As an application, we show that the
characteristic foliation of the discriminantal hypersurface has algebraic
leaves which are either rational curves or smooth elliptic curves.
