  We study the following problem: Given a set $S$ of $n$ points in the plane,
how many edge-disjoint plane straight-line spanning paths of $S$ can one draw?
A well known result is that when the $n$ points are in convex position,
$\lfloor n/2\rfloor$ such paths always exist, but when the points of $S$ are in
general position the only known construction gives rise to two edge-disjoint
plane straight-line spanning paths. In this paper, we show that for any set $S$
of at least ten points, no three of which are collinear, one can draw at least
three edge-disjoint plane straight-line spanning paths of~$S$. Our proof is
based on a structural theorem on halving lines of point configurations and a
strengthening of the theorem about two spanning paths, which we find
interesting in its own right: if $S$ has at least six points, and we prescribe
any two points on the boundary of its convex hull, then the set contains two
edge-disjoint plane spanning paths starting at the prescribed points.
