Small matroid enumeration

On labelled ground sets of sizes $1,2,3$, there are respectively $2,5,16$ matroids, or $2,4,8$ up to matroid isomorphism. At rank of a matroid one, a nonempty subset of nonloops specifies all matroid independent sets of cardinality one. On three labels at rank of a matroid two, the three types are one matroid loop with an independent pair, no matroid loops with one pair of parallel elements of a matroid, and all pairs independent with the triple dependent.