This is an odd problem to have.
I use 'tap to click' usually. This continues to work as it always had. I went to install Mountain Lion, fresh install. This meant I needed to be in the Recovery Mode.
In Recovery, I was unable to select anything using the trackpad at all, although I was able to move the mouse cursor around. Notably, the keyboard shortcuts did not seem to be available.
When I went into my trackpad settings and de-selected 'tap to click' I was unable to click on the trackpad! I had to use the keyboard to reselect it. It is working right now.
Now, I'm certain that this used to work. How can I reset my trackpad to default settings?
EDIT: I also noticed that I can't drag windows anymore. I used to do this by ... well by dragging, holding the trackpad down and moving it. However I just changed to using the 3 finger drag, this works. This started on Lion, and has followed me to Mountain Lion.
EDIT: I'm running a 2009 13" macbook pro unibody. Apparently the keyboard shortcuts, require
<Fn>+<CTRL>+<F2> to get to the Apple Menu, for example. Still, many do not work.