Go to SEB
Today's walking adventure:
Went to SEB (in Tegnérgatan), told I couldn't open an account. Handelsbanken, ditto. Same again at Nordea. Turned up at the Odenplan branch of SEB and opened an account without difficultly.
I opened a private account with a Maestro card without the ability to buy things online (card fee of 15kr per month).
They looked at my passport and work contract. I don't know whether the second was essential. I don't know how this would have played differently if I wasn't an EU citizen.
I never tried Swedbank, I'm still a bit confused by the initial rejection at SEB and things may have changed since August 2016, so please do give answers with your own experiences.