Bicycle and repair shops
Bicycle stores are mostly to be found around Reykjavik. There one can find most spare parts, equipment and general repair service, and even get boxes to pack your bike for the flight home:
Orninn, Skeifunni 11d, 108 Reykjavík. Map
G.A.P. Faxafen 14. 108 Reykjavík. Map
Markid, Ármúla 40, 108 Reykjavík. Map
Hvellur, Smiðjuvegur 30, 200 Kópavogur. Map
Borgarhjól , Hverfisgötu 50, 101 Reykjavík. Map
Hjólið, Smiðjuvegi 9, 200 Kópavogur. Map
Kría cycles , Hólmaslóð 4, 101 Reykjavík. Map
And outside the Reykjavik area:
Sportver, Glerargotu 28, 600 Akureyri. Map
Sportvik, Skidabraut 2, 620 Dalvik. Map
Hjólabaer, Austurvegi 11, 800 Selfoss. Map
One can not expect to find all spare parts or tools in the countryside, so it's advisable to bring your own or buy all the necessities before leaving Reykjavik. It is hard to find bicycle repair shops in the countryside but service stations or garages for cars are to be found in most villages where one can get help (if needed) and repairs.
For a more detailed list check out the Cycling Iceland map 2016