  In this paper we prove that the property of being scattered for a
$\mathbb{F}_q$-linearized polynomial of small $q$-degree over a finite field
$\mathbb{F}_{q^n}$ is unstable, in the sense that, whenever the corresponding
linear set has at least one point of weight larger than one, the polynomial is
far from being scattered. To this aim, we define and investigate $r$-fat
polynomials, a natural generalization of scattered polynomials. An $r$-fat
$\mathbb{F}_q$-linearized polynomial defines a linear set of rank $n$ in the
projective line of order $q^n$ with $r$ points of weight larger than one. When
$r$ equals $1$, the corresponding linear sets are called clubs, and they are
related with a number of remarkable mathematical objects like KM-arcs, group
divisible designs and rank metric codes. Using techniques on algebraic curves
and global function fields, we obtain numerical bounds for $r$ and the
non-existence of exceptional $r$-fat polynomials with $r>0$. In the case $n\leq
4$, we completely determine the spectrum of values of $r$ for which an $r$-fat
polynomial exists. In the case $n=5$, we provide a new family of $1$-fat
polynomials. Furthermore, we determine the values of $r$ for which the
so-called LP-polynomials are $r$-fat.
