  For a hereditary permutation class $\mathcal{C}$, we say that two
permutations $\pi$ and $\sigma$ of $\mathcal{C}$ are Wilf-equivalent in
$\mathcal{C}$, if $\mathcal{C}$ has the same number of permutations avoiding
$\pi$ as those avoiding $\sigma$. We say that a permutation class $\mathcal{C}$
exhibits a Wilf collapse if the number of permutations of size $n$ in
$\mathcal{C}$ is asymptotically larger than the number of Wilf-equivalence
classes formed by these permutations. In this paper, we show that Wilf collapse
is a surprisingly common phenomenon. Among other results, we show that Wilf
collapse occurs in any permutation class with unbounded growth and finitely
many sum-indecomposable permutations. Our proofs are based on encoding the
elements of a permutation class $\mathcal{C}$ as words, and analyzing the
structure of a random permutation in $\mathcal{C}$ using this representation.
