  Packing problems, which ask how to arrange a collection of objects in space
to meet certain criteria, are important in a great many physical and biological
systems, where geometrical arrangements at small scales control behaviour at
larger ones. In many systems there is no single, optimal packing that
dominates, but rather one must understand the entire set of possible packings.
As a step in this direction we enumerate rigid clusters of identical hard
spheres for $n\leq 14$, and clusters with the maximum number of contacts for
$n\leq 19$. A rigid cluster is one that cannot be continuously deformed while
maintaining all contacts. This is a nonlinear notion that arises naturally
because such clusters are the metastable states when the spheres interact with
a short-range potential, as is the case in many nano- or micro-scale systems.
We expect these lists are nearly complete, except for a small number of highly
singular clusters (linearly floppy but nonlinearly rigid.) The data contains
some major geometrical surprises, such as the prevalence of hypostatic
clusters: those with less than the $3n-6$ contacts generically necessary for
rigidity. We discuss these and several other unusual clusters, whose geometries
may shed insight into physical mechanisms, pose mathematical and computational
problems, or bring inspiration for designing new materials.
