Don't know, but we do. We haven't any neighbours to worry about though in the area we shoot - we are surrounded by farmland and have to keep the rats down. I am sure there must be laws in built up areas though.
Try the info on here carrust. Tells you what you can and can't do. Make sure you have decent pellets though - pellets for target practice are no good because they have a rounded end.
And seeing as you want to eradicate the blighters (and they are horrid) - is there any reason they are attracted to your back garden? Ie, nice piles of wood to nest in, vegetables growing, ready supply of food, a pond?
Don't have much luck with rodents barmaid. I have an allotment, & I store my spuds in a shed at the bottom of my garden..I got an infestation a couple of years ago. I got rid of them by setting traps. Last year I got squirrels nesting under my bath. They cost me £90-00 to get rid of. I'm a keen birdwatcher, & I've got loads of feeders scattered round the garden...that's whats attracted them this time.
BTW, thanks for the links.
Like you carrust, I am a keen bird watcher and have bird feeders which attract the rats, so we are always going to have problems. As I said before we are surrounded by farmland and the farm buildings with their barns and grain stores are very near. We also have land drainage ditches front at back. We don't stand a chance. We get rats in our roof from time to time as well because they climb brick and flint buildings very well!!
and what's more the rabbits have eaten the tops of all my emerging crocuses between yesterday afternoon and lunch time today when I went on my tour of inspection!!!