There are several standard constructions of matroids in the literature. We list some below.

The Fano matroid: A matroid which is not representable over the real numbers, but is representable over the field with two elements.

The Hesse matroid: The matroid underlying a Hesse configuration of nine points.

The MacLane matroid: Obtained by deleting any element from the ground set of the Hesse matroid. This matroid is non-orientable.

The NCube matroid: The matroid underlying the vertices of an $n$ dimensional cube.

The NonFano matroid: The matroid obtained by removing one non-basis from the Fano matroid.

The NonPappus matroid: The matroid obtained by removing one non-basis from the Pappus matroid.

The Pappus matroid: The matroid on nine points realizing the collinearities of Pappus' theorem.

The Tic-Tac-Toe matroid: A matroid on nine points whose dual is non-algebraic. It is unknown if the tic-tac-toe matroid is algebraic.

The uniform matroid: A matroid where every $r$ subset of $n$ elements is a basis.

The Vamos matroid: The smallest matroid which is not representable over any field.

$\mathrm{with}\left(\mathrm{Matroids}\right)\:$

$\mathrm{with}\left(\mathrm{ExampleMatroids}\right)\:$

$M≔\mathrm{UniformMatroid}\left(3\,7\right)$

$M≔⟨\mathrm{th\ⅇ\; uniform\; matroi\ⅆ\; of\; rank\; 3\; on\; 7\; \ⅇl\ⅇm\ⅇnts}⟩$

$\mathrm{evalb}\left(\mathrm{numelems}\left(\mathrm{Bases}\left(M\right)\right)=\mathrm{binomial}\left(7\,3\right)\right)$

$\mathrm{true}$

$\mathrm{AreIsomorphic}\left(\mathrm{Deletion}\left(\mathrm{Hesse}\left(\right)\,\left\{1\right\}\right)\,\mathrm{MacLane}\left(\right)\right)$

$\mathrm{true}$

James G. Oxley. Matroid Theory (Oxford Graduate Texts in Mathematics). New York: Oxford University Press. 2006.