Around 50 planetary nebulae (PNs) are presently known to possess ``small-scale'' low-ionization structures (LISs) located inside or outside their main nebular bodies. We consider here the different kinds of LISs (jets, jetlike systems, symmetrical and nonsymmetrical knots) and present a detailed comparison of the existing model predictions with the observational morphological and kinematical properties. We find that nebulae with LISs appear indistinctly spread among all morphological classes of PNs, indicating that the processes leading to the formation of LISs are not necessarily related to those responsible for the asphericity of the large-scale morphological components of PNs. We show that both the observed velocities and locations of most nonsymmetrical systems of LISs can be reasonably well reproduced assuming either fossil condensations originated in the asymptotic giant branch (AGB) wind or in situ instabilities. The jet models proposed to date (hydrodynamical and magnetohydrodynamical interacting winds or accretion disk collimated winds) appear unable to account simultaneously for several key characteristics of the observed high-velocity jets, such as their kinematical ages and the angle between the jet and the symmetry axes of the nebulae. The linear increase in velocity observed in several jets favors magnetohydrodynamical confinement compared to pure hydrodynamical interacting wind models. On the other hand, we find that the formation of jetlike systems characterized by relatively low expansion velocities (similar to those of the main shells of PNs) cannot be explained by any of the existing models. Finally, the knots that appear in symmetrical and opposite pairs of low velocity could be understood as the survival of fossil (symmetrical) condensations formed during the AGB phase or as structures that have experienced substantial slowing down by the ambient medium.