  The zeroes of Goss polynomials $G_{k, \Lambda}(X)$ for $\Lambda = A \defeq
\mathbb{F}_{q}[T]$ and similar lattices $\Lambda$ are studied. Generically, the
zero distribution follows a simple pattern governed by the $q$-adic expansion
of $k-1$. However, if $q=p^{f}$ with $f \geq 2$ is a proper power of the prime
$p$, irregularities of the $q$-adic sum-of-digits function may lead to
deviations from this pattern, to irregular zeroes, and an abundance of trivial
zeroes of $G_{k, \Lambda}$ compared to the generic formula. These phenomena are
related to properties of the Sheats compositions of natural numbers $n$
divisible by $q-1$. Among other things, we give a necessary and sufficient
condition for the existence of irregular zeroes of $G_{k,A}$ and a formula for
the vanishing order of $G_{k,A}$ at $x=0$.
