  Neighborly polytopes are those that maximize the number of faces in each
dimension among all polytopes with the same number of vertices. Despite their
extremal properties they form a surprisingly rich class of polytopes, which has
been widely studied and is the subject of many open problems and conjectures.
  In this paper, we study the enumeration of neighborly polytopes beyond the
cases that have been computed so far. To this end, we enumerate neighborly
oriented matroids --- a combinatorial abstraction of neighborly polytopes ---
of small rank and corank. In particular, if we denote by OM($r,n$) the set of
all oriented matroids of rank $r$ and $n$ elements, we determine all uniform
neighborly oriented matroids in OM($5,\leq 12$), OM($6,\leq 9$), OM($7,\leq
11$) and OM($9,\leq 12$) and all possible face lattices of neighborly oriented
matroids in OM($6,10$) and OM($8,11$). Moreover, we classify all possible face
lattices of uniform $2$-neighborly oriented matroids in OM($7,10$) and
OM($8,11$). Based on the enumeration, we construct many interesting examples
and test open conjectures.
