OK do NOT read this if you are sensitive:
OK. Sigh. My neighbor and I ended up, in our attempts to put down our first trapped possum, effectively doing some testing that PETA would *not* approve of. Our conclusions are that pellets do NOT do it. CB shorts/longs, a type of very low-powered .22 ammunition, do NOT do it. The best round, in terms of effectiveness with least noise, is called the Remington Subsonic, it's actually a "standard" velocity, such as is used in target shooting, with a hollow-point bullet. Shoot this out of a rifle, NOT a pistol since pistols with their short barrels are just loud. You can use a head shot, but a possum skull is not like a persons. Their actual skull is long, narrow, and it's easy for a bullet to glance off. To me the most effective way is to aim for what the deer hunters call the "boiler room", the heart/lung area. For a deer, side on, you're aiming for the spot right behind the point of the shoulder. On a possum, it's the corresponding place. Get an idea of anatomy and visualize how the angles work out. The gun will go bang and you will see some blood. The possum will be in shock pretty much instantly, and die pretty quick. It is about the most humane way to put one down that's available to the average person.
It would be nice if you could just chuck rocks at them and make them go/stay away, or play Barry Manilow music to repel them (although then you'd probably get no eggs, either). But Mother Nature doesn't work that way.
The way I think about it is this: Every time I neat chicken, every time I eat beef, every time I eat lamb or fish, I'm responsible for a killing. Eating meat but not at least mentally taking responsibility for the killing, is the worst way to deal with this issue, yet it's what probably 90% of Americans do. I was telling a friend of ours, a big ol' bruiser of a guy too, how I killed and we ate one of our roosters, and he was amazed. Mr. Macho had never done it (although he's kept chickens for years, apparently his dogs kill enough of 'em that it's all he can do to keep some alive) and wants me to teach him. He was asking me questions, aghast. I think he was beginning to wonder if I was some kind of a monster, and I said, "Look, I've eaten chicken about a million times or so, and loved it every time. If I am going to eat chicken, then I have to take responsibility for what it takes to eat chicken".
This is also why we don't have names for the sheep, the geese, or our chickens. You don't name your food. Human beings have the ability to bond with animals and make them part of our family, or tribe. You don't eat your friends. So, you have to make a distinction between friend-animals, and animals that are pests, or food animals, that you maintain a "professional distance" from.