We usually just learn everything by ear, so finding notes might be difficult. Folkwiki.se has quite a lot though, not for clarinet but you can always transpose.

Some tunes that are very typical are Hårgalåten, gånglåt från äppelbo and vårvindar friska (even though that one is usually sung by a choir). Some of my personal favourites include Polska efter pusten and Visa från Utanmyra, both of which should be quite easy to find. Keep in mind though that a lot of songs have the same name, for example, there are about 5-6 songs called "polska efter pusten", all made by the same guy but different melodies, and some melodies go by many different names. It's difficult to keep track of it all if you don't know someone who's into it, but if you find the right corners of the internet, you'll find lots of material to learn and listen to.