The Preponderence of Evidence is what the law respects, nothing else. No offended sensibilities or whatever we think is "right," matters. So the thing to do is to turn the law to your advantage and have the evidence on your side.
The advice to know your laws in your locale is excellent. They can be widely different from place to place. SO get clarification immediately... knowing them is half the battle. That is your first step.
Any time you even think there's a stray mutt hanging around, call the sheriff. Get a report on file each time and ask for a receipt. Show the deputy the actual law, in writing, whatever it may be.
Be polite, never let them see you sweat and get to know your deputies, in person. You pay their wages, but they are people, too. For all you know, they may wild-eyed dog lovers themselves.
On the other hand, when they believe you empathize with them, well... it cannot hurt.
If you can track down the owner of the dog(s), do not approach him or her in a belligerent way. Speak to them politely - ONCE - and give them a written copy of the law. Then send them a registered confirmation of your discussion and a registered copy of the same to the sheriff, as well. Then call the sheriff personally, ask if your letter was received and let him react in his or her own way to the situation. All of this builds a trail of evidence and draws in the only people who can help, outside of yourself.
Then, get a high powered pellet gun and blast the pooch... often.
I recommend the Benjamin/Sheridan pump models in .22 caliber. They are pricey, but pack a wallop and you can vary the power. They're worth every cent and will last a lifetime. It's highly unlikely the law will frown on an air rifle and everyone ought to have one, anyway.
When you pull the trigger, ensure the animal sees you. In the future, they should associate you and your place with the pain of the shot - pain being all that really deters a stray dog.
(I like the notion of paint ball guns; they point a glaring, accusing finger at the dogs owner. But, I've been shot with PB guns and for all but the smaller dogs, there isn't enough deterrent effect, IMHO.)
Lastly, obtain a humane trap and capture the critter. By then, the sheriff and/or animal control folks should be old pals with you and their resistance to handling the problem should be minimized.
Of course, the final solution is S.S.S. - but not until you have cleared that path, or the dog is caught with a chicken in it's mouth. As it stands right now, you're not there yet.
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Remember, you are in this for the long haul. Be steadfast and determined and dont expect results overnight. In the meanwhile, do like Cyn suggests and become an expert on fences -
and electric fences in particular.
