I'm just using a standard Dreamhost plan. At some point I may have to consider dedicated hosting, but the cost outlay (~$500/month) scares me. And except when something gets submitted to Digg, I usually don't need it. I'll cross my fingers that the more recent versions of phpBB are more secure...
If you want a dedicated hosting plan, look at serverbeach.com. They have plans starting at $75. I have heard great things about them.
I guess the real advantage of a dedicated hosting plan is complete control of your server. If you are doing serious development work or have backup requirements it will make life much easier. Otherwise, a shared plan should be enough.
Btw, phpBB has an IRC chat channel on irc.freenode.net. Great place to ask technical questions about phpBB.