That vine is Smilax and goes by several common names including “Catbrier”. It is difficult to control with herbicides since the foliage is glossy, causing sprays to run off and not be absorbed. It also has underground storage structure which are able to resprout readily.   The most effective option is to dig it up. This may be tedious but over time is the best way to eradicate the plant. If the vines are larger you can cut the vine and “paint” the surface with a product containing triclopyr (Poison Ivy Killer) or glyphosate (Roundup), undiluted. With small, stringy vines this is not practical.