I've started hearing various sources in the last few years advising that you should get the flu shot in your dominant arm, because the more you use the arm, the less soreness it develops. So I've been doing that for the flu shot. Haven't had any issues with soreness, but I'm not sure if I would have anyway.
Then I heard similar advice about the Covid vaccines - that the more you use your arm, the less soreness you'll have. I couldn't bring myself to get it in my dominant arm, though, since I wasn't really sure how much worse than the flu shot it would be. But I did make a point of moving the arm, lifting little hand weights while sitting around, and so on. I would say it was sore "to the touch" for a couple of days but nothing serious. I'm debating whether to get dose #2 in my dominant arm. All in all, I'm not sure any of this makes much difference.