1. In Reykjavik and surroundings

One place is Ishestar (Ice-horses) in Hafnarfjordur.
Another place is Laxnes (close to the house of our Nobel-writer Halldór Laxnes) in Mosfellsbær.
Both of those offer different trips from 1 hour up to day trips and also a combination of horseback-riding and some other trips like a trip to the Blue Lagoon or the Golder circle.
Swimming pools close by are Sudurbaejarlaug in Hafnarfjordur, and
Lagafellslaug in Mosfellsbær.