Links to the open-source self-hosted options (same order as in the poll, no preference implied, but with some comments from my side):
phpBB Lightweight, mature, newer versions can work on mobile, has some interesting but obscure plugins, old-fashioned forum
myBB Mostly the same as phpBB I think, comment below why you would recommend a switch regardless
Kuena Joomla board or similar; would allow additional functionality as part of CMS like Joomla, Drupal or Wordpress (bbPress) etc.
Discourse Modern de-facto standard, Ruby based, a bit of a resource hog, no real sub-forums, only categories
NodeBB Modern & very responsive, NodeJS based, lots of gamification features and real-time chat, apparently somewhat buggy
Flarum Discourse like, but php based / more leightweight, only categories, no stable release yet
Vanilla forums no real idea, but seems popular with gaming communities

I wrote "upgrade" in the poll, where I felt the system was sufficiently technically different to be seen as a major functional change to phpBB.
Please note that some of these options might not allow importing old posts and users from phpBB! The options below would for sure be "restarts".
Some open source Reddit clones I know off (not sure that is really a full replacement for a forum though):
Voten Modern, responsive
Raddle very close to Reddit
FreePost leightweight, clean
Voat very close to Reddit
Edit: Lemmy federated
Some open-source Twitter clones (definitely not a replacement, but maybe sufficient? Could be done in addition, but account sharing is a bit challenging)
Mastodon federated microblogging
GNUSocial (with Quvitter theme?) federated microblogging
Movim more of a chat system with microblogging, but has group chat and comments, federated through XMPP
Some open-source social network sites that include forum like functionality:
Diaspora Federated social network
Hubzilla Federated social network, microblogging and nomadic website tool-kit
Humhub Social network tool-kit, not federated
Elgg Social network tool-kit, not federated
Please comment below with additional suggestions.