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Strange inverses in the group rings of torsion-free groups: https://arxiv.org/abs/2102.11818, via https://www.quantamagazine.org/mathematician-disproves-group-algebra-unit-conjecture-20210412/
This result of Giles Gardam disproves the strongest of the three Kaplansky conjectures on group rings (https://en.wikipedia.org/wiki/Kaplansky%27s_conjectures). See also an interview with Gardam at https://www.uni-muenster.de/MathematicsMuenster/news/artikel/2021-03-04.shtml
It's just an isolated example at this point but it does show that group rings are less well-behaved than had been hoped.
arXiv.orgA counterexample to the unit conjecture for group ringsThe unit conjecture, commonly attributed to Kaplansky, predicts that if $K$
is a field and $G$ is a torsion-free group then the only units of the group
ring $K[G]$ are the trivial units, that is, the non-zero scalar multiples of
group elements. We give a concrete counterexample to this conjecture; the group
is virtually abelian and the field is order two.