This questions seems to come up every time there's an article on HN about the general trend of "Github is the new resume" and I really don't think it's that important, so I wouldn't worry about it. If a company asked you to send them a "Github URL" and you replied that you use Mercurial more, so all your code is on Bitbucket, and here's the URL, and then the company said they weren't interested in Mercurial users, would you really want to work there? (I think Bitbucket hosts Git now anyway so this is almost moot)

I can imagine a clueless recruiter doing that, but an engineer? There's no excuse for it, and I just don't think it's happening. I could be wrong of course, this is just speculation. Honestly, even if you sent a link to a SourceForge SVN or CVS repo (I guess that would be like emailing from a Hotmail address?) you might stand out because still relatively few resumes come attached with a portfolio.

I don't think BitBucket is a disadvantage. It also shouldn't bother if you link to your own hosted public repository when it's easily browsable and readable.

The problem with that is that my code tends to start out a mess -- get the shit done. And then it will be refactored into something that is prettish.

You're not testing enough. TDD and BDD coupled with making sure you follow Agile methods even with "hello world" sample code will prevent this. I would explain more but I am just a follower, A.K.A. a lemming. Got to go fork another RoR project on github. BRB. /s

Probably short. At least in the python world, Bitbucket was very popular until recently - and maybe still so.

GitHub is just shorthand for online open-source project hosting.