While focusing heavily on learning Icelandic, I dug deeply looking for Icelandic-language music. There are surprisingly high amounts of Icelandic bands that sing exclusively in English. Here are my favourite songs with Icelandic lyrics:
Mammút - They play energetic rock music, bordering on post-punk sometimes. The first Icelandic language band I liked, and my favourite still. Here’s Svefnsýkt: Bónus: Rauðilækur
Q4U - Postpunk from the 80s, mostly in Icelandic. They’re still around. Here’s Sigurinn: Bónus: Snjóhvít
Samaris - Atmospheric minimal electro/triphop with clarinet. Very haunting. Here’s Góða Tungl: Bónus: Stofnar Falla
Nóra: Indie rock, ranging from upbeat and energetic to atmospheric and introspective. The more I listen, the more I like it. They sing entirely in Icelandic. Here’s Himinbrim: Bónus: Sjónskekkja
Lockerbie - Post-rock, but with a bit more pop/rock than usually characterised by Icelandic post-rock. Here’s Snjóljón:
Úlfur Úlfur - Hiphop with awesome backing percussion. They sing entirely in Icelandic. Here’s Út: Bónus: Ég er farinn
Pascal Pinon - D.I.Y., lo-fi, indie pop electronic. Many songs in Icelandic. Here’s Ekki Vanmeta: Bónus: Sandur
Sólstafir - Post-black metal, with elements of country and rock. They are amazing live. Their album Svartir Sandar is entirely in Icelandic and highly recommended. Here’s Fjara: Bónus: Svartir Sandar
There’s lots of music that I’ve discovered that I didn’t cover here, but I only wanted to showcase the music I liked the most. Feel free to leave other suggestions in the comments.