I have used SMF and phpbb and never had any stability issues with either.

Can you explain a little more about what you mean by stable?

I second VBulletin. PHP and SMF are good, but VBulleting is the preferred choice amoungst big forum owners.

I would go hands down with Vbulletin as long as you can afford it. This is the most stable and when your forum builds, it can handle the load. It also has the most add ons and skins . If you look at the really large forums Online, they are all using Vbulletin. Just make sure you have a budget to cover it and are going to put a lot of work in building your forum. It is a waste of money to buy the best software if you do not invest in building the forum.

First of all,vbuletin and IPB are the only " paid " and reliable forum software that people now-a-days prefer
But the backup thing ....The forum softwares don't keep backup,it is the host that keeps the backup

So if i were you i would surely go for Vbulletin

The reason i choose vbulletin is because it is user friendly,gives more functionality that IPB and it is way fast than IPB is

So go for Vbulletin

I strongly suggest PHPBB, I have used it on three forums and it has never failed me.