If you ever want to pass it around, or actually see it in print anywhere, though, it would be best to pick either historical characters and situations (e.g. Harry Turtledove's Guns of the South), or literary characters that have passed into the Public Domain, like Dracula (e.g. P.N. Elrod's Quincy Morris, Vampire). There's an almost infinite number of characters and situations a writer can choose from, without stepping on another writer's copyright.
The problem is that things aren't passing into the public domain anymore, and that's across the board. There is a ton of copyright abuse, because there's no incentive not to. And on top of that the extensions to copyright passed in law every time some big corporation realizes one of its cash cows is about to pass into the public domain. As it stands, IIRC, pretty much the entire 20th century is going to be off limits until well into this one.
I honestly don't have a problem with fanfic. It tends to stay off in its own little corners of the internet, not hurting anyone, not making any money off of it, as far as I can see. If you don't want to know about it, you can happily live in ignorance (as I mostly have). Having looked into it a bit more- thanks to tvtropes, I now know a little bit more about fandoms than I did previously (which basically amounted to "they exist"). In some cases, I now wish i could go back ot that state, but anyway- I don't really see the difference between it and, oh...rampant speculation. It's just fan communities having a little fun. How is it different than having a conversation about it? And if anyone claims to have never, ever, once in their life had a conversation along the lines of, "hey, you know that movie franchise we just saw, wouldn't it be cool if in the next one they did XYZ?" I will call them a liar.