  We consider twelve different ways of modelling the 3-body problem in
dimension $\geq 2$. These can be viewed as models of classical and quantum
background independence. We show that a different type of monopole is realized
in each's relational space: a type of reduced configuration space. 8 cases
occur in 2-$d$, and 4 distinct ones in 3-$d$; these reflect counts of
non-equivalent subgroup actions of $S_3 \times C_2$ and $S_3$ respectively. The
$S_3$ acts on particle labels; the extra $C_2$ corresponds to the purely 2-$d$
option of whether or not to identify mirror images. The non-equivalent
realization is due to a suite of subgroup, orbit space and stratification
features.
  Our 2-$d$ monopoles include 4 known ones: a realization of Dirac's monopole
in relational space rather than its more habitual setting of space, the 2-$d$
version of Iwai's monopole, and indistinguishable particle monopoles with and
without mirror image identification. The 4 new ones are indistinguishable under
a 2-particle label switch or under even permutations, in each case with
optional mirror image identification. Our 4 3-$d$ monopoles are 2 known ones:
the actual Iwai monopole and its already-announced indistinguishable-particles
counterpart, and 2 new ones: the two-particle label switch and even permutation
cases. All 4 3-$d$ cases are stratified. The three even-permutation cases are
orbifolds, two with boundary, the 3-$d$ case's boundary constituting a separate
stratum, giving a stratified orbifold. We document each of the 12 cases'
underlying shape space and relational space, and each monopole's Hopf
mathematics, global-section versus topological quantization dichotomy, Dirac
string positioning, and Chern integral concordance with topological
contributions form of Gauss--Bonnet Theorem.
