  In the present work, we revisit two-component Bose-Einstein condensates in
their fully three-dimensional (3d) form. Motivated by earlier studies of
dark-bright solitons in the 1d case, we explore the stability of these
structures in their fully 3d form in two variants. In one the dark soliton is
planar and trapping a planar bright (disk) soliton. In the other case, a dark
spherical shell soliton creates an effective potential in which a bright
spherical shell of atoms is trapped in the second component. We identify these
solutions as numerically exact states (up to a prescribed accuracy) and perform
a Bogolyubov-de Gennes linearization analysis that illustrates that both
structures can be dynamically stable in suitable intervals of sufficiently low
chemical potentials. We corroborate this finding theoretically by analyzing the
stability via degenerate perturbation theory near the linear limit of the
system. When the solitary waves are found to be unstable, we explore their
dynamical evolution via direct numerical simulations which, in turn, reveal
novel waveforms that are more robust. Finally, using the SO$(2)$ symmetry of
the model, we produce multi-dark-bright planar or shell solitons involved in
pairwise oscillatory motion.
