If you had asked this question a month ago my vote would have been "I don't eat bananas". But since I'm on a health kick right now I've been eating a banana a day. And I heard somewhere that if you peel the banana from the bottom you don't get the little stringy things. The first few I tried it held true. However the last few I've had still had some stringy things on it, but not as much as when you peel it from the top.
When I heard about peeling from the bottom the person said to watch monkey's peel a banana. They know how to get it done!