MediaWiki

Mediawiki is definitely the greatest Wiki out there - counted by the number of Wikis on the web, by the amount of content available on different WikiMedia Projects and by the huge traffic they have.
I have been using MediaWiki for my own projects and it's been running well. WikiSPAM happens from time to time, there is AntiSpam extensions, although they could be easier to install.
MediaWiki is most suited for sites which try to collect content and build a knowledge-base similar to wikipedia.
If you need a wiki for project/corporate use, choose one, that has usergroups and permissions integrated. This is possible somehow, but not a priority for Mediawiki developers.

MediaWiki is the leading wiki engine in the world, and many other wiki engines are written under the influence of standards initiatted by MediaWiki.

Unlike wishcow, I think with the recent changes to the MediaWiki core, and addition of WYSIWYG extensions (and a lot more exntesions), MediaWiki is still far beyond its competitors.

There is poor support for rich editors like FCKEdit and TinyMCE, the overall interface is starting to look a bit dated. I would recommend this wiki especially because it is the one used for wikipedia, and thus very recognizable by people and lowers the learning curve for using it.

MediaWiki is undoubtedly the most advanced wiki engine that has been already written; at least I have not found anything similar. It is the best for creating a knowledge base with Wikipedia-like structure and organizing the community content, for example tutorials and articles.

One of the advantages of Mediawiki is very rich set of community extensions that add extra features to MW. Many of them have a very good quality, especially if they are used at Wikipedia. At the second place, I would put the powerful text formatting syntax.

The only quite significant disadvantage is quite ugly source code. I find it annoying especially if I need to customize the default layout for my own needs. Fortunately, many changes can be done directly through editing the special wiki pages that represent CSS styles, system messages etc. - we just have to know, what to change to achieve the requested result.