Two complementary methodologies for extracting useful insights into electronic structure and bonding from contemporary wavefunctions are compared. The first of these, known as the analysis of domain-averaged Fermi holes (DAFH), mostly provides visually appealing descriptions of the role and the extent of electron sharing in chemical bonding. The second one, known as the fragment, atom, localized, delocalized and interatomic (FALDI) charge density decomposition scheme, uses the partitioning of certain localization and delocalization indices to focus on highly visual contributions associated with individual domains and with pairs of domains, respectively. Four variants of a FALDI-like approach are investigated here in some detail, mostly to establish which of them are the most reliable and the most informative. In addition to ‘full’ calculations that use the correlated pair density, the consequences for the DAFH and FALDI-like procedures of using instead a popular one-electron approximation are explored. Additionally, the geometry dependence of the degree of acceptability of the errors that this introduces for delocalization indices is assessed for different formal bond multiplicities. The familiar molecular test systems employed for these various linked investigations are the breaking of the bonds in H2 and in N2, as well as the nature of the bonding in B2H6, as a simple example of multicenter bonding. One of the key outcomes of this study is a clear understanding of how DAFH analysis and a particular variant of FALDI-like analysis could be most profitably deployed to extract complementary insights into more complex and/or controversial bonding situations.