A point: you can't renounce US citizenship if all you have is permanent residency. You have to have citizenship elsewhere before the US will recognize you having renounced its citizenship.
I've seen it happen.
Not that I would recommend it, however.
Source: I'm an international tax lawyer and have 18 expatriation projects open in my office right now.
I personally know someone here in Paraguay who has done it. He's now stateless. The Embassy highly suggests that you have another citizenship. However, they will accept a renunciation if you don't have one.