In unpublished work, Geelen proved that a matroid is near-regular if and only if it has no minor isomorphic to: U2,5; U3,5; the Fano plane and its dual; the non-Fano and its dual; the single-element deletion of AG(2,3), its dual, and the matroid obtained from it with a Delta-Y operation; and P8. We provide a proof of this characterization.
Academic Press
European Journal of Combinatorics
Matroid Structure for Efficiency
Networks and Optimization

Hall, R., Mayhew, D., & van Zwam, S. (2011). The excluded minors for near-regular matroids. European Journal of Combinatorics, 32(6), 802–830. doi:10.1016/j.ejc.2011.01.013