  In this paper we consider $4$-dimensional steady soliton singularity models,
i.e., complete steady gradient Ricci solitons that arise as the rescaled limit
of a finite time singular solution of the Ricci flow on a closed $4$-manifold.
In particular, we study the geometry at infinity of such Ricci solitons under
the assumption that their tangent flow at infinity is the product of
$\mathbb{R}$ with a $3$-dimensional spherical space form. We also classify the
tangent flows at infinity of $4$-dimensional steady soliton singularity models
in general.
