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.