I'm so glad you posted this! I couldn't find it earlier. My neighbor called me yesterday evening. He had trapped a little skunk in his live trap inside his chicken coop and doesn't have a gun.Shooting them with a pellet gun in the live trap is my method. Got rid of 2 this summer. They do spray, but you can stay out of range.
I had no idea what I was getting into, so I took a revolver and a rifle. My Romanian training rifle made me look like a good shot. I had to shoot through two doors, both covered in chicken wire. The little thing had not yet sprayed! The neighbors were able to open both doors for me. I was about 12 feet away. He kept looking at me. I was able to hit him in the head. He immediately turned around! I thought I'd missed but then he flopped over. About a minute later he started smelling like a skunk. I've eaten skunk, but not this one.
We've had them get in the bee hives before, but I've never had the misfortune of catching one before.