From Comparison of Internet forum software (PHP):

Name Creator Current stable version Cost (USD) License Threaded User themes Calendar Image attachment Unread message tracking WYSIWYG Editor PostgreSQL MSSQL Oracle SQLite Firebird
bbPress [1] Automattic 0.8.3.1 Free GPL No No No No No No No No No No No
IceBB [2] XAOS Interactive 1.0-rc8 Free GPL No Yes No Yes Full No No No No No No
IPB [3] Invision Power Services Inc. 2.3.4 $149.99 Standard, $299.99 Business Proprietary Yes Yes Yes Yes Full Yes No Yes ($60) Yes ($60) No No
miniBB [4] Paul Puzyrev & Sergei Larionov 2.1c Free / Paid plugins GPL No Yes Yes Yes ? No No No No No No
MyBB [5] MyBB Group 1.2.12 Free Other Yes Yes Yes Yes Session No No No No No No
phpBB [6] phpBB Group 3.0.0 Free GPL No Yes No Yes Full No Yes Yes Yes Yes Yes
PunBB [7] Rickard Andersson 1.2.16 Free GPL No Yes No No Session No Yes No No Yes No
SMF [8] Simple Machines 1.1.4 Free Other No Yes Yes Yes Full No No No No No No
UseBB [9] Dietrich Moerman 1.0.8 Free GPL No Yes No No Session No No No No No No
Vanilla [10] Mark O'Sullivan 1.1.4 Free GPL No Yes No No Full No No No No No No
vB [11] Jelsoft Enterprises 3.7.0 Beta 3 $85 one yr.+updates, $160 lifetime + $30/year updates Proprietary Yes Yes Yes Yes Full Yes No No No No No
Name Creator Current stable version Cost License Threaded User themes Calendar Image attachment Unread message tracking WYSIWYG Editor PostgreSQL MSSQL Oracle SQLite Firebird

I made this table by stringing together the data from that article about a subset of the listed forum packages, preferring the three most popular ones and any others that were freely available, out of beta (except one that is almost out of beta), with their own Wikipedia articles, and deemed by me to be at least halfway-decent; I omitted the latest release date here, but all of them have had a new release at least in 2007.

I removed some columns of important data because they were all the same in order to save space: All of these forum packages allow flat mode and can use MySQL databases, while none can use flat-file or Access databases.