I'd back pretty much what sv has just written, although I have been out of touch with the UK situation for a long time.
If an animal is classed as 'vermin', then it's literally open season on them, and the restrictions on hunting 'game' don't apply.
Leeds_lobber did mention an incident he had with a duck. Wonder if he can explain further since he seems to have some connections to the legal profession?
Then, of course, laws may differ between England and Scotland
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Given current attitude in the UK, I wouldn't want to start fighting to formally legalise the sling, since it would likely be immediately banned
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The main thing it has going for it is that most people think a sling is a bit of a toy (David and Goliath not excepted), and maybe also that it is hard to get good with it.
As for helping to keep down rabbits with a sling - again I agree with sv. I'd back a man with a dog or a ferret against a large group of slingers any day.