This dataset contains the 385370 pairwise non-isomorphic matroids from 0 to 9 elements inclusive. Each matroid is given as a text string occupying one line of the file matroids09_rankLine. Except for the empty matroid, the text string gives the rank of every subset of the groundset (as described in more detail in the readme.md file), and so is called the rankline of the matroid. This establishes a de-facto numbering scheme for the matroids, starting with M0 (the empty matroid) and ending with M385369 (the uniform matroid U(9,9)). For example, M298 is the number for the Fano plane, and its rankLine occurs on line 298 (counting from 0) of the file. Various published papers in matroid theory have already referred to specific matroids using the M-numbers obtained from this catalogue. This dataset is provided as a "reference version" of the catalogue to permit researchers, both now and into the future, to safely use M-numbers as unique and unchanging identifiers to refer to any matroid on up to 9 elements.