I had a big brambles problem in a flower bed, and just sprayed Roundup on the leaves. This is a systemic weedkiller ( ie it travels down from the leaves and kills the roots) which is also bio-degradable, leaving no residue in the soil.

Once they died back I just dug around the roots and pulled 'em out. If you can't dig out the roots, well just leave 'em.

BUT, if you've got the space somewhere, brambles = BLACKBERRIES!! Yum Yum. Blackberry and apple, from the garden, is the best of home produce IMHO.