Guessing you are in the UK? I am afraid this is a nightmare weed and very difficult to get on top of. It has a waxy covering so weedkiller works better if you can rough it up a bit first. Luckily I have no personal experience eith this but I know people who have conquered it with herbicides, digging and a lot of patience. RHS advice

Hi thanks for the comment, i have been told digging makes the problem spread. unless it is deep excavation and removal

This plant has a very deep and widespreading root system........the plants in your neighbor's garden are related/attached by roots to those in your garden. Unless your neighbor works to eliminate the problem in his garden, you will continue to battle the problem in yours. Digging and removing what roots you can find can help and will not encourage spreading. But removal of ALL top growth as it appears is better. Without an ability to photosynthesize (or produce the spore producing fertile stalks), the roots eventually weaken and die off....at least in that area.

Equisetum has a high silica content and this is what resists herbicides. If you cut the plants back and then apply a strong herbicide (not vinegar - it will have virtually no effect) to the cut stems, it will be translocated down into the root system. But again, this is not a one time shot - you must continue to do this when you see any growing stems.

Persistence is the key here :-) You will never fully succeed in eliminating a plant that has been around for 30 million years but persistence and determination can keep it under control and out of your growing areas.

By digging, I meant really digging, not just pootling about with a fork.