HoneysuckleCredit: morguefilesTrying to dig out wild honeysuckle, smilax or cat's claw, and wild rose vines from around shrubs and hedges, can be a losing battle. There is a way to kill these and other unwanted weedy vines without hurting the plants they are growing through and around and without too much effort. Needed supplies include, a 5 gallon plastic bucket with a lid, some type of strong herbicide labeled for killing woody vines, chemical resistant gloves, protective clothing, and possibly some cutting tools like garden snips or loppers depending on the size of the vine.
The seeds of most weedy vines are spread by birds that eat the vine's fruits. Birds love to nest and perch in bushes and hedges, so that is where the seeds sprout and get a start. Checking around bushes and shrubs will undoubtedly turn up vines that are well rooted under the branches and even up to the main trunk. Left alone, the fast growing vine will cover up the bush and rob it of needed sunlight and the vine's roots will steal the soil's nutrients. The roots of honeysuckle and wild grape will be well entwined with those of the bush making it impossible to dig one out without harming the other. Herbicidal sprays used on the weedy vine can drift or splatter and hurt or kill the nearby bush. 
SmilaxCredit: morguefilesMost of the vine's leaves will be produced at the end where it reaches the topside of the bush and is in the sun. First it is necessary to pull the vine out of the shrub without breaking any connection to the vine's roots. Leave as many of the green leaves on the vine as possible so they will absorb the herbicide. The herbicide will only be absorbed by the green growing leaves, so this method won't work during the winter while the vine is dormant. Trying to handle vines like cat's claw or smilax, and wild rose will definitely require using gloves and wearing long sleeves, due to their thorny stems. With twining vines like honeysuckle that twist around branches, some of those branches may need to be cut off to keep the vine whole.
Take the bucket and fill it half full of water and then add a strong herbicide that is labeled for killing woody vines such as ones that contain glyphosphate or triclopyr. The solution in the bucket needs to be very strong, so add twice as much as the label calls for when making a spray solution. Herbicides are dangerous chemicals be sure to wear protective clothing and to mark the bucket well to show that it contains a herbicide. Be sure the bucket doesn't leak and if the lid doesn't have a hole in it, cut one about two or three inches in diameter.
Wild grape VineCredit: morguefilesFeed the leafy ends of the vine through the hole in the lid of the bucket so that they stick out of the bottom of the lid. Push as many leaves and live vine stems as possible through the lid to be submerged in the herbicide solution. These leaves and green stems will take up the herbicide and feed it directly to the vine's roots killing it. If the vine is very large it may be necessary to cut it off leaving only a four to six foot long section and then when new growth comes out, stick it in a bucket of herbicide. Put the lid on the bucket and let the vine soak for a month or so. Be sure the lid is secure and keep children and pets away from it.
After a month or so, check and see if the vine's stem is dead back to the ground by scratching the bark to see if there is green tissue beneath it. If the stem is dead, cut it off and remove the dead vine ends from the bucket while still wearing gloves. Whatever herbicide solution is left can be used to pour over other unwanted weeds and brush. It will have probably lost most of it killing power but having been out of the sun it may still be quite lethal. If the vine has sprouted around its base, use the solution to treat those sprouts, just use an old paintbrush to apply the herbicide. Often the vines will resprout some distance from the treated stem in an attempt to regrow. Smilax, with its large root system will especially do this and may require several treatments. This treatment can be used to kill any type of weedy vine that is actively growing without endangering wanted shrubs and bushes.