Yes, vBulletin is secure. This forum and many other large forums use vBulletin. http://www.vbulletin.com/forum/showthread.php?194701-How-To-Make-My-Forums-More-Secure is a nice thread on how to improve and increase the level of security with your vBulletin installation.
As for now,i think it is most secure in the lot. The closest competitor is phpbb but phpbb being free ,may be may not hire people to look constantly into securities,features as rapidly as vbulletin. After all they can use that money and i feel vbulletin is bit expensive as well.
vBulletin is very expensive but there price worth there script. Its very secure and many options to manage your forums through Admin Panel. Yes CyberJ37 is right, beware from modifications you don't believe. Most hackers are only seeking the holes through these modifications. Anyways, vBulletin has great library of professional templates too. Good Luck
It's the most expensive software i know of, but it's very secure and you shouldn't have any security issues, and if you do they'll get fixed pretty quickly.
The ultimate answer is that: vBulletin is as secure as much as how secure your server is; with no modification.
Most of the modification in vBulletin.org are template edits, though some has database edits but still safe.
Most - but not all. I remember a website found a security vulnerability in a mod I was using. No one informed either myself or the mod author - I only came across it in a search for my website :x Not the most pleasant thing. I prefer to have less mods, as it makes migration, upgrading etc. easier and less time consuming -> wasting less money.