  Recent experimental studies of $^{87}$Rb spinor Bose Einstein condensates
have shown the existence of a magnetic field driven quantum phase transition
accompanied by structure formation on the ferromagnetic side of the transition.
In this theoretical study we examine the dynamics of the unstable modes
following the transition taking into account the effects of the trap,
non-linearities, finite temperature and dipole-dipole interactions. Starting
from an initial state which includes quantum fluctuations, we attempt to make
quantitative comparisons with recent experimental data on this system and
estimate the contribution of quantum zero-point fluctuations to the domain
formation. Finally, using the strong anisotropy of the trap, we propose ways to
observe directly the effects of dipole-dipole interactions on the spinor
condensate dynamics.
