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One of the most recent encounters I had where multipoles were important was the electric dipole moment (EDM) of the electron.
The current upper bound for the EDM of an electron sits 9 orders of magnitude above the prediction of the standard model. This is why researchers are looking at systems where these EDM’s are enhanced. They do this to try and see if the prediction of the standard model is correct, or whether there is other physics at work. One example of such a system is nuclei that exhibit octupole deformation.
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