My driving instructor was absolutely brilliant - Bob Brown, 01235 532676. He does Oxford and Abingdon, and is the gentlest, most patient, relaxed teacher imaginable: really good at building up your confidence as well as your skills. I passed first time. Good luck!

I passed with Botley Driver Training. Tel: 07821 164850 Really friendly.

I passed fourth time so am probably not much help! However I did Pass Plus with Bob Brown, and would also endorse him very highly.
Personally I would say that the best independent driving instructors are generally better than large companies like BSM, but of course the independent people probably also vary more. But in general the people who have their names on their roofs have more at stake!
You're always likely to find the best people by asking for personal recommendations.

I'd avoid BSM:
http://www.guardian.co.uk/money/2011/apr/16/bsm-instructor-drove-me-round-in-circles

Thanks for all the replies- very helpful -