Photobucket is unsuitable for animated GIF's, especially if they're looping. Not many days after you've posted, you'll get a "Zemg your bandwidth is expiring" email, followed by your images being unviewable. Unless you pay for the upgrade, which wouldn't be worth it if you ask me. Of course, you need to have more than one GIF active on the internet, but for those of us who can't be assed to track them down and delete them, this is an issue. It does have the advantage of keeping your uploads in an accesible area, so you don't have to upload them again, though, which can be pretty neat at times.
ImageShack/TinyPic (they're pretty similar) doesn't have the bandwidth problem, and the upload limit is pretty decent. ImageShack has a 1.5 MB limit on images (it will optimize them if they're larger), while TinyPic can host bigger files, albeit with a 1600 pixel size restriction (resizes if larger) on both width and height. None of them save your images in an album unless you make an account, of which i'm pretty sure is free. There's no limit to how many images you can upload (without account, don't know about with), but i've read that TinyPic deletes your images after 90 days of not being viewed if you don't have an account.
So yeah, that's pretty much it. I personally use the latter two.