Small remarkable matroids

Question

I'm working on a problem involving matroids $M=(E,\mathfrak{C})$ (here $E$ is the ground set, $\mathfrak{C}$ the set of circuits) with a "small" ground set $E,$ in the sense that

$\sharp(E)\leq7$

I want to concrete compute some examples. I've already reach some results in the case of uniform matroids.

However, I'm interested in the study of some cases of non uniform matroids over a ground set $E$ with $\sharp(E)\leq7.$ Clearly, I know there exists a library with the enumeration of all possible matroids. The question is the following:

Are there some remarkable* examples of such matroids?

Thank you very much for every helpful suggestion.

*In the sense that people are more interested (for geometric/historic/something else reason) in such examples rather that others?

Answer 1

A matroid is called regular if it is representable over all fields. The Fano matroid $F_7$ is a matroid with 7 elements which is a forbidden minor for regular matroids. The other forbidden minors for regular matroids are the uniform matroid $U_4^2$ and $F_7^*$ the dual of the Fano matroid.