I don't know how to make this community wiki. Anyway, what I am looking for are things in physics that offend mathematicians because of the lack of rigor. I would also like answers that mention things that were non-rigorous in the past, even if they now have a rigorous formulation or definition.
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List questions, especially those without any underlying conceptual question, are generally considered off-topic here. Beyond that, there is no real answer to this question. With the exception of overly-excitable math students, I suspect that most people understand that full rigor is neither necessary nor even desirable for most physics applications, and that there is an entire branch of physics dedicated to worrying about the technicalities which most others gloss over. Due to their efforts, pretty much everything I can think of [...] – J. Murray Apr 20 '21 at 23:37
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[...] which is used in physics has a relatively rigorous formulation somewhere. Things that don't - path integrals, etc - are active areas of research for both communities. – J. Murray Apr 20 '21 at 23:39
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See my answers to Angular velocity of precession, or Why is force a vector?. I talk about this issue. – mmesser314 Apr 21 '21 at 02:07