The Unit Distance Problem asks:
For a set of $n$ points in the plane, what is the maximal number $g(n)$ of unit distances realized among the ${n \choose 2}$ pairs?
A properly scaled square grid gives a lower bound of something like $g(n) \ge n^{1 + \frac{c}{\log \log{n}}}$, and a beautiful application of the crossing number lemma gives that $g(n) = O(n^{4/3})$.
A closely related problem where great progress was made very recently is the Distinct Distance problem, asking for the minimum number $f(n)$ of distinct distances among $n$ points in the plane. (Clearly $f(n)g(n) \ge {n \choose 2}$.)
Guth and Katz recently obtained a sharp exponent for $g(n)$. f(n)$. Terence Tao and János Pach wrote nice summaries of this work.
