Special and simplest coordinates for a vector field in a neighborhood of a point

Question

If $v$ is a smooth vector field in a smooth manifold $M$, there always exist a chart over any $p \in M$ such that, in a neighborhood of $p$, $v$ has coordinates $v^1=1$ and $v^j = 0$, for all $j=2,...,n$ in this chart, or it's just true when $p$ is a regular point of the vector field(i.e.: $p$ is a point such that $v_p \neq 0$)?