  Observing constituent particles with fractional quantum numbers in confined
and deconfined states is an interesting and challenging problem in quantum
many-body physics. Here we further explore a computational scheme [Y. Tang and
A. W. Sandvik, Phys. Rev. Lett. {\bf 107}, 157201 (2011)] based on valence-bond
quantum Monte Carlo simulations of quantum spin systems. Using several
different one-dimensional models, we characterize $S=1/2$ spinon excitations
using the spinon size and confinement length (the size of a bound state). The
spinons have finite size in valence-bond-solid states, infinite size in the
critical region, and become ill-defined in the N\'eel state. We also verify
that pairs of spinons are deconfined in these uniform spin chains but become
confined upon introducing a pattern of alternating coupling strengths
(dimerization) or coupling two chains (forming a ladder). In the dimerized
system an individual spinon can be small when the confinement length is
large---this is the case when the imposed dimerization is weak but the ground
state of the corresponding uniform chain is a spontaneously formed
valence-bond-solid (where the spinons are deconfined). Based on our numerical
results, we argue that the situation $\lambda \ll \Lambda$ is associated with
weak repulsive short-range spinon-spinon interactions. In principle both the
length-scales can be individually tuned from small to infinite (with $\lambda
\le \Lambda$) by varying model parameters. In the ladder system the two lengths
are always similar, and this is the case also in the dimerized systems when the
corresponding uniform chain is in the critical phase. In these systems the
effective spinon-spinon interactions are purely attractive and there is only a
single large length scale close to criticality, which is reflected in the
standard spin correlations as well as in the spinon characteristics.
