Spray any existing sticker bushes with a premixed glyphosate herbicide. Direct the herbicide only at the sticker bushes, as it will kill any plant it contacts. Spray on a calm day so wind doesn't carry the herbicide to other plants.
Pull up the dead sticker bushes after a few days, getting as much of the root as possible. Wear work gloves so you aren't poked by the thorns.
Cut out the grass under the fence using a spade. Remove any grass or weeds along the fence line and at least 6 inches away from the fence to prevent new weed growth near the structure.
Dig down about 3 to 4 inches in the area where you removed the grass to make a trench for mulch. The mulch will suppress any future weed growth.
Place landscape fabric in the trench. Secure it in place with landscape staples.
Pour a heavy mulch or filler, such as pea gravel, into the trench. The weight of the material will keep it from blowing away. Spread the material to create an even layer under and around the fence area.