Is there a solid "Bitcoin Bank" yet?
Currently, the most secure place to store bitcoins online appears to be the exchanges. There is some evidence that Mt. Gox is also hoping to become the default bitcoin bank as they are developing mobile payment applications for instant payment.

http://mtgoxlive.com/mobile/

They have also reimbursed the losses of another bitcoin exchange.

https://www.mtgox.com/press_release_20110811.html

It is worth pointing out that while Mt Gox was hacked, no one lost any bitcoins that were not reimbursed.