  We begin by revisiting a paper of Erd\H{o}s and Fishburn, which posed the
following question: given $k\in \mathbb{N}$, what is the maximum number of
points in a plane that determine at most $k$ distinct distances, and can such
optimal configurations be classified? We rigorously verify claims made in
remarks in that paper, including the fact that the vertices of a regular
polygon, with or without an additional point at the center, cannot form an
optimal configuration for any $k\geq 7$. Further, we investigate configurations
in both triangular and rectangular lattices studied by Erd\H{o}s and Fishburn.
We collect a large amount of data related to these and other configurations,
some of which correct errors in the original paper, and we use that data and
additional analysis to provide explanations and make conjectures.
