One Family Dental, 1 Miller Street, Morningside - I have been a patient there and am happy with them. It takes NHS patients and still had space on its books for new patients the last time I heard,
and Dr Tooth, aka Fairmilehead Dental Practice, 6 Oxgangs Road (not far from Redford and Dreghorn) http://www.drtooth.co.uk/
I know them but have never been treated there. A friend of mine, however, whose father is was a tutor at a dental school tells me the senior partner there is reputed to be the best in edinburgh.