A foldable pocket saw, clippers, gloves, and a long-handled lopper are the basic tools. Prybars pull large English ivy vines off a tree's trunk so they can be cut. 

An electric chain saw hits big vines and bushes. The hedge trimmer soars through heavy infestations of multiflora rose and masses of small vines such as kudzu. 

Herbicide use should minimal but at times is essential. Its effective for infestations of porcelain berry seedlings and multiflora rose. A quart sprayer drips super-concentrate glyphosate on fresh cut stumps of vines and plants to kill roots. 

English ivy forms massive carpets that slowly kill trees and native undergrowth. SEE https://hgic.clemson.edu/factsheet/english-ivy-control/

Expose all the vines with a downward sweep of your pocket saw to remove leaves. Pry vines from the tree trunk to cut. Crowbar helps. Vines form a network so all must be cut. 

Cut and remove sections of vines close to ground. Small amounts of concentrated Roundup (glyphosate) dripped on fresh cut vine stumps stop regrowth. Even better, pull vines up by roots.

Porcelain Berry forms great hanging curtains that slowly kill trees and smother ground vegetation. 

Porcelain berry looks very similar to native grape vines but berries are distinct and vines have a white core while grape is dark brown. 

Cut hanging vines at the base and head high to clear trees. Large vine roots are killed with super concentrated Roundup dripped on the fresh cut vine stump. 

Vines produce thousands of seedling. They and ground cover readily killed when point sprayed with triclopyr. 

Is extremely hard to remove due to its extraordinarily robust root system.

 https://www.fs.usda.gov/Internet/FSE_DOCUMENTS/fseprd529922.pdf

Do not mow! Small fragments of the stem can grow into new plants. 

Grows extremely fast as it smothers trees and ground cover. 

Spray ground foliage with glyphosate or triclopyr 

Kudzu is not easy to remove, but persistent attack will kill it.  

This immense spread of kudzu is basically one plant that has spread with lengthy, quick growing runners. A tap root can produce 20 runners many yards long. 

Cut Kudzu vines and runners at the base root and "juice." 

Rose stems branching from the base grow into immense, dense thickets of sharp thorns. 

These thorns are really nasty. 

Small plants filling a woodland are removed by spot spraying with triclopyr.

For large plants, cut back branches with a hedge trimmer and then "juice" the stumps. Watch out for those thorns!