Joe's plan = This is fine. We don't know whether Joe has one plan or several. All we know is that this particular plan was thought up by Joe. A plan of Joe's = This is also fine, but it gives us more information. It tells you that Joe has a number of different plans, and this is only one of them. A plan of Joe = This is wrong. We can't say this.

I'd prefer "Joe's plan". After that, "a plan of Joe's" works, although it's a little colloquial and possibly regional. It works best when "plans that Joe has" are a reasonably usual occurrence. You can think of it as short for "a plan of Joe's devising." "A plan of Joe" is technically correct but it unusual, jarring, and (to some) unpleasant to read. Btw, "a cup of Joe" means "a cup of coffee" so that might be why I find that third option so odd. :)