I would definitely not recommend phpBB.
It's the most popular free forum software available, but it's buggy, insecure and often the target of hacking.
For free forum software I'd recommend MyBB. We use this and it's incredibly feature rich, and much better written than phpBB.
Another good alternative would be SMF.
For paid software, vBulletin and IPB were the common choice.
UKBF used to run on vBulletin I believe, and now runs on XenForo.
XenForo is modern and lightweight, and normally the go to choice for paid forum software now.
To throw something different into the mix, you could look at discourse.org or nodebb.org
Things to look out for are how resource intensive the software is - e.g. is it lightweight, its anti spam capabilities (either natively, or through plugins), its feature set (do you need particular features), and whether you need it to connect to another system e.g. for shared logins with a WordPress website.
If you're planning to have a standalone forum, I wouldn't recommend bbPress. I wouldn't even recommend it if you were running WordPress to be honest.
Softaculous's demos and reviews can be useful:
https://www.softaculous.com/softaculous/apps/forums
This is generally regarded as the best one click installer, which you can expect any decent host to run.
MyBB and SMF are the highest rated on there.
If you want a forum that'll hold up under high load and lot of users, use XenForo or SMF.
If you want a forum that's feature rich, use vBulletin, IPB or MyBB.
I hope that helps.
Why not install a few and give them a try?
All the best,
Dan