I'm reasonably happy with SEB so far. I originally tried to get an account with SwedBank but they weren't interested until I had a personnummer (legally they're not supposed to refuse on this basis!) SEB were a lot more cooperative and set me up with a basic account (deposit for salary checks plus an ATM card) quickly. Most things seem to be fine through the website.

The one time I had to do some non-standard stuff they were pretty slow and a little disorganised, but it still seemed better than the UK banks I've used previously.