Ivy - if growing up trees or shrubs needs to be cut off at the bottom of the tree, it will be easier to pull off when it has died back. Bramble, ivy etc - strim it to ground level, then keep mowing or strimming until it goes. If you can mow it - the grass will gradually take over. Otherwise strim to ground then cover securely with thick carpet (wool only) or membrane (like Mypex) make sure you secure the edges. It will always take time - even if you use chemicals you will probably have to treat it several times to get rid.

SBK brushwood killer applied to thick roots - see my blog on tree stump treatments, it also applies to ivies and brambles - you need to take down the top growth, dig around and reveal the hard, woody roots (and with brambles, the central woody lump of root) and treat as described.

A few years ago we had Ivy growing into our dry stone wall. When we gained access to the other side of the wall, the trunk was almost 7" in diameter. I teased what I could out of the wall & hammered a packet (200 I think) of copper nails into the remaining 'branches'.
I hacked through the trunk, drilled some holes into the stump & poured in root killing crystals, plus some more copper nails for good measure. It hasn't returned!!
We have brambles come through from the car park at the bottom & I've given up. I've never been able to control it up until last year, when in desparation, I gave it a sturdy support to grow up. Seems that it's happier to go up than spread out, & the few tendrils that did stray were easily taken care of.