Here is a list of all matroids on at most 9 elements as well as all 2-polymatroids on at most 7 elements, up to isomorphism. They are given as a list of flats and their ranks. The flats themselves are encoded as bitmasks. For example, $U_{3,4}$, the uniform matroid of rank 3 on 4 elements, is stored as:
0,0 1,1 2,1 4,1 8,1 3,2 5,2 6,2 9,2 a,2 c,2 f,3
The trailing spaces on each line are left there intentionally.
The larger files are compressed with bzip2. Use of wget is suggested.

Matroids on 0 elements
Matroids on 1 element
Matroids on 2 elements
Matroids on 3 elements
Matroids on 4 elements
Matroids on 5 elements
Matroids on 6 elements
Matroids on 7 elements
Matroids on 8 elements
Matroids on 9 elements

2-Polymatroids on 0 elements
2-Polymatroids on 1 element
2-Polymatroids on 2 elements
2-Polymatroids on 3 elements
2-Polymatroids on 4 elements
2-Polymatroids on 5 elements
2-Polymatroids on 6 elements
Due to limitations of this webhost, the 2-polymatroids on 7 elements are stored in 11 different files. When uncompressed, these take up 92 gigabytes of space. I have made no effort to economize the storage.
2-Polymatroids on 7 elements, part 1
2-Polymatroids on 7 elements, part 2
2-Polymatroids on 7 elements, part 3
2-Polymatroids on 7 elements, part 4
2-Polymatroids on 7 elements, part 5
2-Polymatroids on 7 elements, part 6
2-Polymatroids on 7 elements, part 7
2-Polymatroids on 7 elements, part 8
2-Polymatroids on 7 elements, part 9
2-Polymatroids on 7 elements, part 10
2-Polymatroids on 7 elements, part 11

The algorithm to construct the catalog of 2-polymatroids is described in the following paper:
Savitsky, Thomas J., Enumeration of of 2-polymatroids on up to seven elements, SIAM J. Discrete Math., 28(4) (2014), 1641--1650. doi

My source code that constructs that catalog may be downloaded here: kpolym.
Unfortunately, the kpolym program is neither user-friendly nor terribly well documented; it also contains other functionality. Compiling it would require the source code to nauty as well as a working installation of igraph and ATLAS. For parallel execution, I used autoson; see this link for a quickstart guide to autoson.

Also, I have written software to test the k-base-orderability of matroids. The source is here: kbo. The notion of k-base-orderability is discussed in the following paper:
Bonin, Joseph E. and Savitsky, Thomas J., An infinite family of excluded minors for strong base-orderability, Linear Algebra and its Applications, 488 (2016), 396-429. doi

I have also created a rudimentary C library so that k-base-orderability can be checked via the matroids package for SageMath, but this code is not (yet) available here.
I have only attempted to compile any of this source code on Ubuntu Linux.

If you are actually interested in using these programs, let me know, and I will spend time cleaning up the source and creating more documentation. I have placed the source code here mostly as evidence that these programs in fact exist. The current source could obviously use some tidying-up.

Author: Thomas J. Savitsky