We don't have a picket fence, but some of our fences are rather uneven, and the chickens still go on/over them.
My Americauna hens in particular really get up high. I had one on the roof of our house the other day! They roost on the free swinging long lights in my shop at night, must be 8+ feet up off the ground. Darn birds!
The rest of my flock roosts on a low wooden fence in my front yard. I've seen them sitting on everything except the barbed wire and the chain link. They will fly over the chain link fence, but they don't like to stand on it. I've seen them land, wobble, and then quickly fly off.
My garden has 6' woven field fence with a strand of barbed wire over that. Keeps out the deer and the chickens!
Liz