This paper studies the probability of substructures occurring in random Latin squares. Our main result states that if $α,β>0$ are such that $2α+β<1$, then there are positive constants $δ= δ(α, β)$ and $Δ= Δ(α, β)$ such that if $P$ is a partial Latin square of order $n$ with $k = k(n)$ non-empty cells occupying at most $αn$ rows and $βn$ columns, the probability that a random Latin square of order $n$ contains $P$ lies between $(δ/n)^k$ and $(Δ/n)^k$. We apply this result to subsquares in random Latin squares to obtain the first proof of the fact that the expected number of subsquares of order $3$ in a random Latin square of order $n$ is non-vanishing as $n \to \infty$. We are also able to provide the best known asymptotics for the expected number of subsquares of order $a$ in a random Latin square of order $n$ when $2<a=o(n^{1/2})$. Finally, we discuss the implications of our result on other configurations in random Latin squares as well as on completions of partial Latin squares.