Let $(M,\Delta,g)$ be a sub-Riemannian manifold and $x_0\in M$. Assuming that Chow's condition holds and that $M$ endowed with the sub-Riemannian distance is complete, we prove that there exists a dense subset $N_1$ of $M$ such that for every point $x$ of $N_1$, there is a unique minimizing path steering $x_0$ to $x$, this trajectory admitting a normal extremal lift. If the distribution $\Delta$ is everywhere of corank one, we prove the existence of a subset $N_2$ of $M$ of full Lebesgue measure such that for every point $x$ of $N_2$, there exists a minimizing path steering $x_0$ to $x$ which admits a normal extremal lift, is nonsingular, and the point $x$ is not conjugate to $x_0$. In particular, the image of the sub-Riemannian exponential mapping is dense in $M$, and in the case of corank one is of full Lebesgue measure in $M$.