Singular matroid realization spaces

A matroid is realizable if we can obtain its bases from the indices of linearly independent columns of some matrix. For a given matroid $M$, this matrix is not unique. The space of all such matrices can be given the structure of an affine scheme, known as the realization space of $M$. It is known that representation spaces of matroids can be arbitrarily singular, although there are few concrete examples. We use software to study smoothness and irreducibility of representation spaces of rank 3 and rank 4 matroids, isolating examples of singular spaces for $(3,12)$-matroids. As an application, we show that singular initial degenerations exist for the $(3,12)$-Grassmannian.