  As a vector version of scalar Bessel beams, Poincar\'{e}-Bessel beams (PBBs)
have attracted a great deal of attention due to the presence of polarization
singularities and their nondiffraction and self-healing properties. Previous
studies of PBBs have been restricted primarily to understanding the
disinclination patterns in the spatially variable polarization, and many of the
properties of PBBs remain unexplored. Here, we present a theoretical and
experimental study of the polarization characteristics of PBBs, investigating a
variety of their features. Using a mode transformation of a full Poincar\'{e}
(FP) beam in a rectangular basis, ideally carrying 100$\%$ polarization
coverage of polarization states represented on the surface of the Poincar\'{e}
sphere, we observe the PBB as the superposition of an infinite number of FP
beams, as each ring of PBB has polarization coverage >75$\%$. We also observe
the resilience of a PBB's degree of polarization to perturbation. The
polarization-ellipse orientation map of PBBs shows the presence of infinite
series of C-point singularity pairs. The number of such series pairs is decided
by the number of C-point singularity pairs of the FP beam. The dynamics of
C-point singularity pairs in the self-healing process show a non-trivial
creation of new singularities and recombination of existing singularities. Such
dynamics provide insight into ``Hilbert Hotel'' style evolution of
singularities in light beams. The present study can be useful for imaging in
the presence of depolarizing surroundings, studying turbulent atmospheric
channels, and exploring the rich mathematical concepts of transfinite numbers.
