  A classical conjecture by Graham Higman states that the number of conjugacy
classes of $U_n(q)$, the group of upper triangular $n\times n$ matrices over
$\mathbb{F}_q$, is polynomial in $q$, for all $n$. In this paper we present
both positive and negative evidence, verifying the conjecture for $n\le 16$,
and suggesting that it probably fails for $n\ge 59$. The tools are both
theoretical and computational. We introduce a new framework for testing
Higman's conjecture, which involves recurrence relations for the number of
conjugacy classed of \emph{pattern groups}. These relations are proved by the
\emph{orbit method} for finite nilpotent groups. Other applications are also
discussed.
