Animals escaped their pens, in multiple computers and Java versions. It happened with cobblestone and wood fences. There was no fence gate at the time. This happened with chickens, sheep, cows, mooshrooms, and pigs. I did catch them once, but didn't get a snapshot, sorry!

1. Make a fence around an animal, cobblestone or wood.
2. Wait a while. It took a few Minecraft days for it to happen to me.
3. Laugh or watch in horror (depending on the person) as your specified precious animal walks away. Or just dead pan through it. I don't know.