Brambles are blinking awful things, even the dead ones have great big thorns that scratch you to bits. I am also fighting the bramble & bindweed problem, there are quite a few of us on here fighting the battle, and I often come here for advice. Brambles only tend to go down a spade and a half's worth, so double digging an area would get rid of most of it. However, it regenerates from the tiniest bit left, so I would suggest blitzing the area with roundup, sbk brushwood killer or any other weedkiller containing glyphosate. The initial results are dismaying - it seems to green up and grow before your eyes, but does eventually die. Spray, leave 4-6 weeks for it to be absorbed into the root system, then chop down the top growth and dig out the roots (leave a good 6 inches of 'stump' so you can see where you need to dig). Invest in a pair of decent leather gauntlets, it will make your job so much easier (I bought some of the Tough Touch ladies' gauntlets, worth every penny, cheapest are Amazon).
I'm not patient enough for the glyphosate-only route, and as I can only clear at the weekends (hubby around to look after sproglets), what I tend to do is chop down as far as I can reach, chop up and put in green bin, and dig out roots as I go. When I've had enough, the next bit I can see gets sprayed with the roundup, as even if it doesn't kill it, it weakens it.
I also have bindweed, that goes down to the centre of the earth and is impossible to dig out. I pop a cane in, let it twine around that, make a hole in the bottom of a pop bottle that the cane will fit through, and when the bindweed reaches the top of the cane, very carefully slide the cane out through the pop bottle. When all the bindweed is in the bottle - make sure it doesn't snap off, especially near the bottom, screw the top onto the bottle, and spray lots and lots and lots of weedkiller into the bottle (so it's almost swimming in the stuff), and leave. It is slowly getting the message. I think when I've won my battle, I'm going to hire myself out as a bramble clearer, it's a very therapeautic pastime, seeing a virtual jungle transformed into a blank canvas. That point for me, is some way off though!