OWS (Sapphire) at present.
Dropped from GGL, so Gold / Emerald previously (if I tried to add a Finnair cabin bag on a superlight fare with that status, there was no charge and it wouldn't let me complete the transaction).

Added: just had a quick look at the Finnair system and it mentions I have a COAB on my booking (although that vanished and few days later)

Additional summing up:

As OWS / One World Sapphire / BA Silver, on AY metal:

Can select any seat but an exit row in economy
Can select any business seat, normal or the preferred window ones

When I was OWE / One World Emerald / BA Gold / GGL, I could select any seat including exit rows, but could not select the premium economy seats on an A350 (on flights where the cabin wasn't sold as PE) without a £60+ charge.