What are the rules for use of contractions?

I am a native English speaker. I've noticed that there seem to be very strong rules about when to use contractions, but I haven't seen these rules enumerated anywhere. For instance, I think nearly every native speaker would say this is wrong.

"Who's going to the movies?"
"I'm".

But is there a rule that you can't have a one word sentence be a contraction? No. E.g.

"What's your best advice on whether I get insurance from FlyByNite.com?"
"Don't"

Is OK.

There seem to be quite a few of these rules. Are they listed somewhere?

It's OK to contract an auxillary verb (have, has, will, am, is, are, was, were) to its subject if the main verb exists in the sentence. Otherwise, don't do it.

In the first example, the contraction either should not occur or you need to specify the main verb. Answering "I'm going" is OK. Answering "I am" is OK. Answering just "I'm" makes it sound like you haven't finished the sentence.

In the second example, not is not an auxillary verb (it's an adverb modifying imperative "do"), so it's fine.

However - going back to the first example, the rule doesn't apply if not is involved. So you can answer "I'm not." to your first example.