Usability of a static property/method?

From what I understand, a lot of times when people use static methods and classes, it's really just a strict adherence to the OOP paradigm.

For example, an often given example of 'why use static' is the Math class in Java, where all the constants of the class static properties. Of course, you wouldn't want to have to make a new Math object when you wanted to use a constant, so of course you'd want it static....

The less non-trivial counterargument to that being... well, do I really want to encapsulate these values in a class in the first place if my language of choice doesn't require me to? This ties you to the class, and gets dicey when you start looking at using polymorphism to swap classes out and stuff, or so I've come to read.

However, there are times when they seem to be very much useful, the big example that I've had presented to me learning PHP from a variety of sources is using the Singleton pattern of object instantiation in user logins. There's a wiki piece about Singletons here: http://en.wikipedia.org/wiki/Singleton_pattern

Actually, one of the sites I had bookmarked from when I looked into same thing seems to have a pretty good (Java oriented) discussion of this whole thing here (actually it seems to have ended being the primary thing that has informed me on the topic, looking at it some more): http://www.coderanch.com/t/407852/java/java/static-methods