Would this be bounded Let $a_{m}$ be supremum of the minimum of the angle between the line
segments between any $m$ points, in which the supremum is taken over
all configurations of $m$ points. Is $\sqrt{m}a_{m}$ bounded as
$m\rightarrow\infty$ ?

In $\mathbb{R}^2$ it seems unlikely that $a_m\sqrt{m}$ is bounded.
It is known that there is some $c>0$ such that
the number of distinct angles in $[0,\pi)$ determined by $3$-point subsets
of any set of $m$ non-collinear points
is 
at least $cm$.
It is conjectured that the minimum number of angles determined by such a point set is in fact
$m-2$.
The number of distinct directions determined by $m$ non-collinear points in the plane is $m$ or $m-1$ depending on whether $m$ is even or odd.
See section 6.2 of Research problems in discrete geometry.
