What is the most popular design methodology?
I cannot tell which methodology of design is the most popular. However, I can speak from my personal experience. At work, we follow Agile software development practices. The whole Agile philosophy is heavily inclined on frequent feedback and iterative development. For instance, pair programming, a programming technique used by Agile, is an idea where two people code on the problem on the same computer. Typically, one will be at the keyboard typing out code and the other will keep a close eye on the code being produced by his buddy. This way the person coding gets immediate feedback from the other person watching him code. Another benefit is it makes programming more interactive and interesting compared to programming solo.
In the previous paragraph I have talked only about coding, which is a small phase in SDLC. Agile practices can be followed during all phases of SDLC. We actually tell our client that we follow Agile methodlogies and hence expect there constant feedback. Clearly there is so much to Agile that I don't know and that I can't type in this textbox.
And what book would be a good place to start for someone who wants to
learn that methodology.
There are many good books on this topic. One of them is Agile Samurai. I suggest you also read Mythical Man Month and TDD by example. These are the books highly recommended by my own colleagues that I plan on reading myself.
Also, is there a website that offers sample requirements documents?
Yes. The first thing that comes to my mind is a series of articles by Joel Spolsky.