We study some aspects of the dynamics of an analytic vector field $X$ in a neighbourhood of an invariant non-singular curve $\Gamma$ in $\mathbb{R}^3$. Namely, the spiraling behaviour: any trajectory of $X$ {\em spirals\/} asymptotically around $\Gamma$. This is measured by means of the angle, $\theta$, and the distance, $r$, functions of the trajectories with respect to cylindrical coordinates around $\Gamma$. Coordinates are called balanced if these functions are monotone, which is not an intrinsic property. Balanced coordinates always exist in the case of {\em elementary singularity\/} (non-nilpotent linear part) and we show, in the general case when $\Gamma$ is not contained in the singular locus of $X$, the existence of coordinates for which the angle is monotone. These are obtained as {\em maximal contact\/} coordinates for the reduction of the singularity. The results can be viewed as generalizations of the corresponding results in dimension two which we study first as a motivation.