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