We investigate minimization problems along a singular trajectory of a single-input affine control system with constraint on the control, and then as an application of a sub-Riemannian system of rank 2. Under generic assumptions we get necessary and sufficient conditions for optimality of such a singular trajectory. Moreover we describe precisely the contact of the accessibility sets at time $T$ with the singular direction. As a consequence we obtain in sub-Riemannian geometry a new splitting-up of the sphere near an abnormal minimizer $\gamma$ into two sectors, bordered by the first Pontryagin's cone along $\gamma$, called the $L^\infty$-sector and the $L^2$-sector.