Colin C.
Google
Came here as a tourist and would just say - it kicks the pants off pretty much any public pool I have been to in North America. There is one surprising reason for this.
Is there an attendant that makes sure you shower before you go in? Yes! They put these signs everywhere in North America but nobody obeys them. But there is a reason those signs exist. If everyone cleaned off properly, pools wouldn't need to use so much chlorine or bromine to kill off the microbes. Another reviewer said Laugardalslaug uses more chlorine than most pools in Iceland. I have no reason to doubt this, but even then I couldn't detect it in my nostrils afterward.
Bonus (not the supermarket): having a changeroom where the floor is actually dry, because again - they MAKE you dry off before leaving the showers and entering the changeroom. I'm willing to bet athlete's foot is less of an issue here than in North America.
Anyway, if you are in Reykjavik (especially if you are camped next door), do yourself a favour and go to Laugardalslaug. Do the second cheapest thermal cycle you'll likely ever experience (cheapest being the Reykjadalur thermal river). The pools are nice and clean and not crowded. We had a nice chat with some friendly locals in the hot tub. Get a hot dog afterward, the stand outside offers veggie dogs too. Be respectful and follow the rules at the pool. You will thank yourself, and become an evangelist for pre-pool showers. Unfortunately I fear most of my fellow North Americans are too prudish to take them seriously.