The fractions and dimension of bars, rings and lenses are studied in the near-infrared (IR) S0 galaxy survey, which is a sample of similar to 200 early-type disc galaxies, mainly S0s. We find evidence that multiple lenses in some barred S0s are related to bar resonances in a similar manner as the inner and outer rings, for which the outer/inner length ratio is similar to 2. Inner lenses in the non-barred galaxies normalized to galaxy diameter are clearly smaller than those in the barred systems. Interestingly, these small lenses in the non-barred galaxies have similar sizes as barlenses (lens-like structures embedded in a bar), and therefore might actually be barlenses in former barred galaxies, in which the outer, more elongated bar component has been destroyed. We also find that fully developed inner lenses are on average a factor of 1.3 larger than bars, whereas inner rings have similar sizes as bars. The fraction of inner lenses is found to be constant in all family classes (A, AB, B). Nuclear bars appear most frequently among the weakly barred (AB) galaxies, which are consistent with the theoretical models by Maciejewski & Athanassoula. Similar-sized bars as the nuclear bars were detected in seven 'non-barred' S0s. Galaxy luminosity does not uniquely define the sizes of bars or bar-related structures, neither is there any upper limit in galaxy luminosity for bar formation. Although all the family classes cover the same range of galaxy luminosity, the non-barred (A) galaxies are on average 0.6 mag brighter than the strongly barred (B) systems. Overall, our results are consistent with the idea that bars play an important role in the formation of the structure components of galaxies. The fact that multiple lenses are common in S0s, and that at least the inner lenses can have very old stellar populations, implies that the last destructive merger or major gas accretion event, must have taken place at a fairly high redshift.