While WikiIndex lists scores of wiki engines, I wanted to narrow down the list of wiki engines to those that have widespread use or prominent implementations.
All of the wiki engines listed below except for Everything2 are free and open source
|Wiki Engine||Description||Top Wiki|
|CMS related to TikiWiki; written in PHP; modular; high traffic, custom web development.||VOIP-Info|
|Clearspace||Jive Software; written in Java;||BBC newsforums|
|CLiki||Written in Common Lisp;||TUNES Wiki|
|Corendal||Corendal; written in Java; no wiki syntax to learn, a WYSIWYG rich text editor is used instead||collaborative|
|DokuWiki||Splitbrain Software; written in PHP;||Romapedia|
|Everything2||Custom Wiki for Everything Development Company; programming language probably Perl||Everything2|
|JAMWiki||Written in Java; feature parity with MediaWiki||OLAT|
|MediaWiki||Open source; most widely used; developed specifically for Wikipedia. Written in PHP;||1911Encyclopedia|
|MoinMoin||Written in Python;||Edubuntu|
|PmWiki||Written in PHP; mostly used for non-reference sites;||CenterForestResearch|
|PukiWiki||Japanese; written in PHP;||Mostly Japanese|
|Xwiki||Written in Java; enterprise wiki used mainly in France||collaborative|
BitWeaver and MediaWiki are database-oriented; the other wiki engines listed above are file-oriented. The difference is that, with a file-oriented wiki engine, each entry is its own file. In contrast, with a database, each entry is a record in a table belonging to the backend.
I was interested in some of the various alternatives, one of which was the commercial wiki engine Clearspace ($29/user). There I was surprised to learn that the BBC website is (evidently) powered by this product. Clearspace is written in Java, which is very interesting to me for several reasons I'll discuss later. Another product that I thought looks attractive is Corendal.
SOURCES & ADDITIONAL READING: Gmap Package, Bitweaver organization; Wiki Popularity results, Wiki Creole; Comparison of WikiEngines (results for this table), WikiMatrix;