  Investigations of spin squeezing in ensembles of quantum particles have been
limited primarily to a subspace of spin fluctuations and a single spatial mode
in high-spin and spatially extended ensembles. Here, we show that a wider range
of spin-squeezing is attainable in ensembles of high-spin atoms, characterized
by sub-quantum-limited fluctuations in several independent planes of
spin-fluctuation observables. Further, considering the quantum dynamics of an
$f=1$ ferromagnetic spinor Bose-Einstein condensate, we demonstrate
theoretically that a high degree of spin squeezing is attained in multiple
spatial modes of a spatially extended quantum field, and that such squeezing
can be extracted from spatially resolved measurements of magnetization and
nematicity, i.e.\ the vector and quadrupole magnetic moments, of the quantum
gas. Taking into account several experimental limitations, we predict that the
variance of the atomic magnetization and nematicity may be reduced as far as 20
dB below the standard quantum limits.
