  A generalization of the scattering equations on $X(2,n)$, the configuration
space of $n$ points on $\mathbb{CP}^1$, to higher dimensional projective spaces
was recently introduced by Early, Guevara, Mizera, and one of the authors. One
of the new features in $X(k,n)$ with $k>2$ is the presence of both regular and
singular solutions in a soft limit. In this work we study soft limits in
$X(3,7)$, $X(4,7)$, $X(3,8)$ and $X(5,8)$, find all singular solutions, and
show their geometrical configurations. More explicitly, for $X(3,7)$ and
$X(4,7)$ we find $180$ and $120$ singular solutions which when added to the
known number of regular solutions both give rise to $1\, 272$ solutions as it
is expected since $X(3,7)\sim X(4,7)$. Likewise, for $X(3,8)$ and $X(5,8)$ we
find $59\, 640$ and $58\, 800$ singular solutions which when added to the
regular solutions both give rise to $188\, 112$ solutions. We also propose a
classification of all configurations that can support singular solutions for
general $X(k,n)$ and comment on their contribution to soft expansions of
generalized biadjoint amplitudes.
