  We show that if a homogeneous polynomial $f$ in $n$ variables has Waring rank
$n+1$, then the corresponding projective hypersurface $f=0$ has at most
isolated singularities, and the type of these singularities is completely
determined by the combinatorics of a hyperplane arrangement naturally
associated with the Waring decomposition of $f$. We also discuss the relation
between the Waring rank and the type of singularities on a plane curve, when
this curve is defined by the suspension of a binary form, or when the Waring
rank is 5.
