Chopped down about a dozen small vinegar trees (Staghorn Sumac), in the garden, the stumps are at most, 10cm diameter. The roots have really kicked off developing shoots everywhere that come through the grass, grow really fast and are a nightmare to control. Its like a survival instinct as a result of taking the tree down.

Done the stump drilling weed killer trick… no change.

Not a lot on the web for this, but if I do nothing, I will end up with a new plantation of the stuff as its growing so fast…. Is the only answer to dig all the stumps out ?

Explosives.. you need explosives.

Fighting the same (losing) battle here. I stupidly allowed some shoots to get established and like you they’re now everywhere. Glyphosphate weed killer kills the shoots, but takes a week or two, don’t know yet if it gets rid of the underground suckers, but I’m not hopeful.
If you can get the stumps out, I’d go for that…in my case a small nuclear strike is probably needed 🙂

Thinking a Nuke From Orbit is the way to go.

The roots that I have dug up by hand are small underground root balls that go to thin runners quickly. I pull these up like varicose veins across the lawn.

Its spread into the neighbours fields too so want to get on top of it. Never seen anything take over so fast.

Think I need to hire a mini digger for the 20 or so stumps.

Sumac requires a scorched earth policy. Every single little bit of root you leave in the ground will grow into a tiny tree. It is very determined.

If I keep pulling the shoots out, will it give up after a bit of time, or is it almost indestructible ?

Root removal the only option ?

It’s a battle of attrition. The remaining roots are using stored reserves to send up re-growth. They can’t go on forever. If you’ve removed or killed the stumps it helps. Keep taking off re-growth as fast as. Oh, and check in hidden corners, behind garages and such. You’ll be surprised at how far roots spread.

Ammonium sulphamate is the way to go:
http://mistralni.co.uk/products/ammonium-sulphamate?gclid=CIa_0KCM5b8CFQjItAodw1UA4g
Will kill it down to the bottom of the roots and then turn it into compost. Result x 2!
Check out the history of this chemical on Wiki for further info.