Personally I love Bitbucket. Unfortunately you'll get scoffed at by the "true nerds" for using such a second rate product (aka less trendy). Their new UI is really awesome and you really can't beat the price (free).
As a bitbucket user I can say that I've been using it for about 14 months and I can only recall one outage. But take that info with a grain of salt.
It may not be github's software as much as they are the number 1 git hosting service and therefore they are the biggest target for hackers.
I might be talking out of my ass here but it feels a bit like the argument for using Apple versus MS, i.e. there are no viruses for Macs since they are more secure... until Apple grew in popularity, which then attracted more hackers to write viruses for Apple's OS, which showed that it is was not invulnerable.
I doubt Bitbucket will supplant Github as the goto git hosting service, but if they grow they might be subject to more attacks? Just a thought to consider.
I use Github professionally, and for my open source projects. The visibility of a project on Github is hard to beat, and is often found just because it's on Github. Outages usually come on the front-end, and often I've been able to push to Github within minutes of finding an outage. That said, we're a small company with almost no remote collaboration on a given project, so outage doesn't affect us much.
However, for my serious, closed-source projects (that I might want to become products) I use Bitbucket. So far I've experienced no outages.