We present an extensive numerical study of spin quadrupolar correlations in single and coupled bilinear-biquadratic spin one chains, using several methods such as exact diagonalization, density matrix renormalization group, and strong coupling series expansions. For the single chain, we clarify the dominant correlation function in the enigmatic gapless period-three phase for $\ensuremath{\theta}∊(\ensuremath{\pi}∕4,\ensuremath{\pi}∕2)$, which is of spin quadrupolar nature with a period three spatial structure. Then we revisit the open problem of the possible existence of a ferroquadrupolar phase between the dimerized and the ferromagnetic phases. Although an extended critical region is in principle compatible with the numerical results, a scenario with a huge crossover scale is more plausible. Finally we study the fate of the dimerized phase upon coupling two chains in a ladder geometry. The dimerized phase rapidly vanishes and an extended gapped phase takes over. This gapped phase presumably has dominant short-ranged ferroquadrupolar correlations for $\ensuremath{\theta}∊(\ensuremath{-}3\ensuremath{\pi},4,\ensuremath{-}\ensuremath{\pi}∕2)$ and---surprisingly---seems to be adiabatically connected to the plaquette single solid phase of the Heisenberg
S=1 ladder and therefore also with the Haldane phase of isolated chains.
Finite-cell calculations (up to
N=12 spins) have been performed on the spin-1 Heisenberg-Ising chain with an uniaxial anisotropy, $\mathcal{H}=\ensuremath{\Sigma}{i}^{}[{S}_{i}^{x}{S}_{i+1}^{x}+{S}_{i}^{y}{S}_{i+1}^{y}+\ensuremath{\lambda}{S}_{i}^{z}{S}_{i+1}^{z}+D{({S}_{i}^{z})}^{2}]$. From a scaling analysis of the gap between the ground state and the first excited state, a phase diagram has been drawn in the ($\ensuremath{\lambda},D$) plane and the transition lines between the "ferromagnetic," $X\ensuremath{-}Y$," "singlet-ground-state," and "antiferromagnetic" phases have been estimated for the infinite-
N system. One of the most important results is that a singlet-ground-state phase with a nonzero gap exists in an extended range of $\ensuremath{\lambda}$ and
D values including the Heisenberg point $\ensuremath{\lambda}=1$,
D=0, in contrast with the spin-
21 case. Moreover, for $\ensuremath{\lambda}\ensuremath{\simeq}1$, the gap decreases with increasing positive anisotropy
D, goes through a minimum, estimated to be zero, and then increases with
D.