Well, the short answer to this question is that there is no real stand-out solution that excels in all situations.
Prototype is the most comprehensively documented -- albeit in a splintered way. It is also seemingly the most widespread library at the moment, possibly because it really excels at the kind of tasks developers complete most often, like selecting nodes and working with lists. Of course, it's going to be the natural choice for Ruby developers because it sticks to many Ruby idioms. One other great thing about Prototype is that it has the mighty weight of Rails behind it and, as a result, there are many developers providing bug fixes and patches to Prototype. Finally, it offers a wealth of add-on libraries such as scriptaculous, Rico, and Behaviour that make it a good, solid choice for many developers.
On the other hand, Prototype has a very under-developed event handling framework, which is a major problem for an otherwise powerful library. Also -- and this is purely a matter of taste -- Prototype's super-pragmatic approach to things (like the heavy use of the innerHTML property) can seem a little "dirty" sometimes.
For smaller projects, the decoupled design and fully-featured components of YUI may well be a big plus. It's very easy to drop in the Connection Manager or the Event library and get going on some basic tasks without having to traverse too much of a learning curve. On the whole, though, it doesn't have much to offer in terms of cool or powerful features.
In my opinion, Mochikit is by far the most well designed and well thought out of the four, and Python/Twisted/Nevow developers will definitely find its API very familiar. However, its documentation is a bit thin in some places (for instance, I'm still a little unsure as to which version of the Mochikit distribution to put in the script tag). Also, some of the idioms and functional techniques that it uses may be confusing for beginners or those who aren't well versed in functional programming techniques. However, it really is worth a look. Mochikits's capabilities will probably surprise you -- the createDOM function, iteration tools and the asynchronous architecture are a work of art.