  In 1984, Kelly and Oxley introduced the model of a random representable
matroid $M[A_n]$ corresponding to a random matrix $A_n \in \mathbb{F}_q^{m(n)
\times n}$, whose entries are drawn independently and uniformly from
$\mathbb{F}_q$. Whereas properties such as rank, connectivity, and circuit size
have been well-studied, forbidden minors have not yet been analyzed. Here, we
investigate the asymptotic probability as $n \to \infty$ that a fixed
$\mathbb{F}_q$-representable matroid $M$ is a minor of $M[A_n]$. (We always
assume $m(n) \geq \text{rank}(M)$ for all sufficiently large $n$, otherwise $M$
can never be a minor of the corresponding $M[A_n]$.) When $M$ is free, we show
that $M$ is asymptotically almost surely (a.a.s.) a minor of $M[A_n]$. When $M$
is not free, we show a phase transition: $M$ is a.a.s. a minor if $n - m(n) \to
\infty$, but is a.a.s. not if $m(n) - n \to \infty$. In the more general
settings of $m \leq n$ and $m > n$, we give lower and upper bounds,
respectively, on both the asymptotic and non-asymptotic probability that $M$ is
a minor of $M[A_n]$. The tools we develop to analyze matroid operations and
minors of random matroids may be of independent interest.
  Our results directly imply that $M[A_n]$ is a.a.s. not contained in any
proper, minor-closed class $\mathcal{M}$ of $\mathbb{F}_q$-representable
matroids, provided: (i) $n - m(n) \to \infty$, and (ii) $m(n)$ is at least the
minimum rank of any $\mathbb{F}_q$-representable forbidden minor of
$\mathcal{M}$, for all sufficiently large $n$. As an application, this shows
that graphic matroids are a vanishing subset of linear matroids, in a sense
made precise in the paper. Our results provide an approach for applying the
rich theory around matroid minors to the less-studied field of random matroids.
