  We prove limit laws for the number of occurrences of a pattern on the fringe
of a ranked tree-child network which is picked uniformly at random. Our results
extend the limit law for cherries proved by Bienvenu et al. (2022). For
patterns of height $1$ and $2$, we show that they either occur frequently (mean
is asymptotically linear and limit law is normal) or sporadically (mean is
asymptotically constant and limit law is Poisson) or not all (mean tends to $0$
and limit law is degenerate). We expect that these are the only possible limit
laws for any fringe pattern.
