Rational quartic spectrahedra

Rational quartic spectrahedra in $3$-space are semialgebraic convex subsets $\mathbb{R} ^3$ of semidefinite, real symmetric $(4 \times 4)$-matrices, whose boundary admits a rational parameterization. The Zariski closure $\mathbb{C}\mathbb{P} the spectrahedron is complex symmetroid. We give necessary conditions on configurations singularities corresponding symmetroids \mathbb{P} spectrahedra. provide an almost exhaustive list examples realizing configurations, and conjecture that missing example does not occur.