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Bónus and Krónan sell themselves as the cheapest, with Bónus the cheaper of the two.

there's also Kostur, but I've never been there, and some household goods for sale in ellingsen too (canned goods, soap etc).

if you're serious about finding bargains, you should make it a habit to read the newspaper (fréttablaðið). They do regular price-comparison surveys. Today, for example, they have an article about where to buy cheapest fish:



I don't have any more personal experience to offer since my shopping is more focused on finding fresh variety of vegetables, and things that seem to be hard to find in Iceland (tofu, black beans).

The last survey about where is cheaper to shop,made Kronan the winner, but Bonus was just behind.
In Kostur is ok to buy some American stuff (bigger packages of Cheerios and large bottles of soap-softener for the laundry and cleaning stuff); but the local stuff is not that cheap.
Noatun,Hagkaup and 10/11 are just to buy there the few things you don´t find elsewhere.
If you have to buy fish and you don´t mind to get it frozen (like in most of shops are sold),then i recommend you to wait for the weekends and go to the flea market "Kolaportið",where you´ll find more to choose (Saturdays and Sundays from 11h to 17h).
For green stuff and some local stuff, try here:
   Use googletranslator or tradukka.com to translate it  XD
They do have sometimes very good offers and have season products all the time.
There are other supermarkets (Netto,Fjardarkaup...) but prices are not that interesting except some offer they might do.
Hope my indications helped you a bit  ^_^

You are very welcome!
I understand; then,i think your best options are Bonus and Kronan