  A $k \times n$ partial Latin rectangle is \textit{$C$-sparse} if the number
of nonempty entries in each row and column is at most $C$ and each symbol is
used at most $C$ times. We prove that the probability a uniformly random $k
\times n$ Latin rectangle, where $k < (1/2 - \alpha)n$, contains a $\beta
n$-sparse partial Latin rectangle with $\ell$ nonempty entries is $(\frac{1 \pm
\varepsilon}{n})^\ell$ for sufficiently large $n$ and sufficiently small
$\beta$. Using this result, we prove that a uniformly random order-$n$ Latin
square asymptotically almost surely has no Latin subsquare of order greater
than $c\sqrt{n\log n}$ for an absolute constant $c$.
