The problem of state equivalence of a given nonlinear system to a triangular form is considered here. The solution of this problem has been known for the regular case, i.e. when there exists a certain nested sequence of regular and involutive distributions. It is also known that in this case the corresponding system is linearizable using a smooth coordinate change and static state feedback. This paper deals with the singular case, i.e. when the nested sequence of involutive distributions of the system contains singular distributions. Special attention is paid to the so-called bijective triangular form. Geometric, coordinates-free criteria for the solution of the above problem as well as constructive, verifiable procedures are given. These results are used to obtain a result in the nonsmooth stabilization problem.