In 1980, Seymour completely classified regular matroids. In this thesis, we use this classification to provide a complete list of maximal simple regular matroids up through rank six. In each rank, exactly one of these is graphic. Additionally, there are one, two, and eight cographic, but not graphic, maximal simple regular matroids in ranks four, five, and six, respectively. In rank five, we also have the 10-element sporadic matroid, and in rank six we have a 12-element maximal simple regular matroid and 16-element maximal simple regular matroid, both of which are decomposable matroids that are not graphic, cographic, or sporadic. In particular, rank six is the first rank we see decomposable matroids that are not also graphic or cographic. In addition, rank six has a maximal simple cographic matroid that is not maximal simple regular.
