Need to get the roots out as well. Just cutting off at ground level makes them multiply. If digging out is too hard then look at a chemical solution but first you must identify the culprit then check against what the chemical destroys.

Here is a link that might be useful: Propagating Perennials

Sounds like could be bullbrier vine.

Thanks for the tips. I did some research on the bullbrier vine and I dont think thats it. Bullbrier vines have leaves and berries and the thorny vines that I have dont have anything but long growing vines with thorns.