  Enumeration of all combinatorial types of point configurations and polytopes
is a fundamental problem in combinatorial geometry. Although many studies have
been done, most of them are for 2-dimensional and non-degenerate cases.
  Finschi and Fukuda (2001) published the first database of oriented matroids
including degenerate (i.e. non-uniform) ones and of higher ranks. In this
paper, we investigate algorithmic ways to classify them in terms of
realizability, although the underlying decision problem of realizability
checking is NP-hard. As an application, we determine all possible combinatorial
types (including degenerate ones) of 3-dimensional configurations of 8 points,
2-dimensional configurations of 9 points and 5-dimensional configurations of 9
points. We could also determine all possible combinatorial types of 5-polytopes
with 9 vertices.
