I find tagging really attractive as an alternative to strictly hierarchical categories. My main uses of tagging are in del.icio.us, LibraryThing, and Lightroom. Picasa, that I used to use before Lightroom, also has a form of tagging, but it never seemed very "natural" to use it there, probably because it's not very well implemented. Flickr is another big user of tagging, but for some reason I've never gotten into using Flickr much.
Tagging, especially in sites like del.icio.us and Flickr, is often promoted as a "social" tool. I've never really used it that way, maybe because I'm not much of a "social" person :-) Of course, I've benefited to some extent from other people's tagging, but that's never been my main benefit. I primarily use del.icio.us to maintain my personal bookmarks.
As far as implementation, del.icio.us has a nice system, especially for web based software. It shows you all your tags, auto-suggests as you type, shows recommended tags, and popular tags. I have wondered why they "hide" the popular tags (ones other people have used) down at the bottom of the page. I guess this is to avoid the problem of people just applying the same tags as everyone else, which would somewhat defeat the purpose of having a wide variety of people applying tags.
LibraryThing's tagging could definitely be improved. You often have to type tags just from memory, with no auto-completion or suggestions. A system more like del.icio.us would be a lot nicer.
Lightroom has auto-completion and lets you drag and drop tags, but its unique feature (at least, I haven't seen it anywhere else) is its "implied tags". A common issue with tagging is what level of detail to tag at, and how many tags to apply. For example, do I tag with "Saskatoon", or "Saskatchewan", or "Canada", or "North America" or several of these. Similarly, do I tag with "pelican", "birds", or "animals". Ideally (for searching) you'd apply all the relevant tags, but that would make tagging a tedious process. Lightroom lets you create tags as children of other tags, so "Saskatoon" is a child of "Saskatchewan" which is a child of "Canada". You manually apply the most specific tag, e.g. "Saskatoon" and the parent tags are automatically "implied". So if I search for "Saskatchewan", I'll automatically get anything I tagged with "Saskatoon". Very nice. I would like the same feature in del.icio.us and LibraryThing.
On this same topic, I listened to a Long Now talk by Clay Shirky on my run this morning. He was talking about the problems of hierarchical categorization, how it is mainly a result of having to organize physical materials, and that tagging is better suited for digital information.
Also on basically the same topic, I am currently reading Everything is Miscellaneous by David Weinberger (who also co-wrote The Cluetrain Manifesto). It makes many of the same points.
So it seems a little incongruous that a hierarchical system can make tagging better. Of course, in a way it's just a shortcut for entering tags, it doesn't really alter the tagging. And I wonder if the hierarchical part is really essential. At first when I saw "implied tags" in Lightroom, I wasn't thinking of a hierarchy, I just thought that one tag could imply others. But that could be tricky to handle since it would allow loops and if not used "properly" could end up with a mess. But it would be interesting to try.
I keep thinking there should be some way to apply tagging to our business software, but so far I haven't come up with anything really appealing. We could allow tagging things like equipment, but I'm not sure whether that would be a big benefit. Maybe we should try it and see whether people like it. Maybe we could replace things like "Type" and "Role" with more general tagging.