Dreghorn Woods is 'hidden' between Redford Road and the Bypass. It surrounds the Braid Burn and has some great old trees and interesting curiosities such as Salmonberries (flower).
The woodland is still owned by the MoD (it's not that long since it had "Beware Guard Dogs" signs) and 'managed' by the council. The main path can be very muddy, though at this time of year it can be frozen so more passable.
It can be reached from south Morningside mostly on paths (though some diversions are currently required due to the flood prevention works). Route on EveryTrail.
The south side of the wood (downhill path) can be reached from Dreghorn Loan (don't let "Private Road" sign put you off).