I've know people who cut down the plants and dug the roots up every year for 10 years or more...about the time they finally thought they had found every last one, new plants would pop up 30' feet away. Wisteria can do the same sort of thing. Much depends on what sort of soil you have.It is a lot easier to dig it out of sandy loam than some other soils. It also depends on the trumpet creeper. It is easier to dig out 'Madam Galens', for example, than to dig out the native trumpet creeper. When we planted it, I made sure I bought and planted 'Madam Galens' and 'Flava'. Digging and tilling can work if you like to dig and till repeatedly, but often doesn't work in compacted clay or caliche clay soils because you cannot get dig deeply enough to reach all the roots. Or, if you are not physically able to dig and rototill that much, then digging isn't a practical solution...like it isn't for people with bad backs. I dug out lots of stuff when I was younger that I just don't think I'd spend time digging out today. This likely explains why I have wild blackberries and wild grapes growing in the veggie garden where I did not plant them and don't especially want them. Susan, A jackhammer? lol lol lol That might scare away the kitties, hummingbirds and butterflies, and the neighbors likely wouldn't love it either. I haven't tried 20% horticultural vinegar on Trumpet Creeper and wonder if it might work. I think the most effective herbicide would be Tordon or something else labeled as a brush killer. The chemicals in brush killers are stronger than the standard homeowner type herbicides. We used some of the standard herbicides in a futile attempt to kill greenbrier and poison ivy when we bought this land in the 1990s. It was a total waste of time and effort. The poison ivy and greenbrier are still with us to this date, though I am pretty successful at keeping about 90% of the poison ivy out of the fenced veggie garden. It always lurks nearby in the adjacent woodland and attempts to creep into the garden in summer when it is so hot that I just venture into the garden to harvest and then run back to the air-conditioned house. I got the poison ivy out of the garden by digging it out every year, often multiple times per year. Luckily, the poison ivy is at the sandy end of the garden. If it was at the clay end, it would be there still. Dawn

There are at least two plants in the photo but one is obviously English ivy, Hedera helix. It is an invasive plant in a number of areas (especially here in my area of the PNW). It is difficult to remove from surfaces, as it attaches itself with very persistent little rootlets and it does provide a habituate for unwanted wildlife as well as holding moisture against the underlying surfaces. Because ivy has such a glossy, waxy surface to the foliage, most herbicides cannot penetrate to do their damage. Manual removal is the most effective control but you must get all the roots as well......small portions of root left in the ground can sprout new growth very easily. Here is a link that might be useful: English ivy removal

If english ivy is coming in from a neighboring property you just have to keep pulling it up. If the english ivy is solely within your yard you can eventually eradicate it by continued pulling. Pile the vines off of the ground and let them dry out and die. English Ivy growing up trees, etc. should be clipped low to the ground and once more a few feet up and the cut section removed. The remainder above the cut can and should be left to die in place.

the only "common" herbicide that can behave that way is imazapyr (i.e. move from roots to adjacent roots) I've done something similar many times with poison ivy or other vines with either glyphosate or triclopyr (or 3 way) and the non-target species were fine. Once I just painted a huge, hairy 4" PI trunk with 3 way mixed with a bit of dish soap, and it died completely. This was after chainsawing the top of it while wearing a full respirator and eye protection. (with the chain motion spinning away, of course) I then cleaned the chain by rinsing some LV naphtha over it, into my home heating oil tank! Where the concentrated uroshiol all got burned, I hope...anyhow I never got any form of the rash or lung irritation. There's one mistake to your approach. 1/2 minute could work with the right herbicide and surfactant, but much better is to weigh down a container and stick the main stem in there for a full 24 hours. Or if you worry about leaving it overnight because of some miscreant coming by and knocking it over, at least a few hours while you are on-site. This summer I killed some horrid Smilax vines by dipping the overturned stems in a test tube of conc. 41% roundup during a dry spell, and taping it back to the base of the vine. Because it was dry they drank it all up overnight.