  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 theta in
(pi/4,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 theta in (-3pi,4,-pi/2) and -- suprisingly --
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.
