Thanks for all your replies. I had been using unfiltered, heated tap water (20 C), and I think this is the problem. I just learned that the apartment building I'm living in has had some upgrades to the water pipes. Today I saw tiiiny little particles in the heated tap water. I processed a film today, and filtered this (heated) water through a coffee filter before mixing it with the developer. Stop bath and the first couple of rinses were done with unfiltered heated tap water with particles, but I gradually cooled the film down and ended up using cold water for the final rinse. Couldn't see any particles in this cold water, and the film turned out just fine. No spots at all. I ran a squeege over the negative as well without problems. A bit silly introducing the squeeging as a second variable, though. Now I don't really know whether it was the use of cold water for the final rinse or the squeeging that did the trick, but I really suspect it was the water. This was just a quick test roll from a camera I might buy, so I didn't bother getting hold of any wetting agents before developing. I'm a bit more relaxed now, but maybe I'll buy some next time I'm in the shop.
For the next time, I'll pour some cold tap water in clean bottles, let it heat up to room temperature and use that water for the developer + final rinse. Thanks for all your clues about how to resolve this problem.