While going through the 'Introduction to Maya 2013' course, I ran into an issue while editing polygons by moving vertices.
I accidently pressed something on my keyboard which meant that when I moved the vertices (in X, Y and Z), it curved the whole line (like a smooth curve) rather than pivoting from the next vertex in a straight line. Not sure how to undo this (whether I need to go into settings or it's a simple keyboard shortcut) and it's driving me nuts.
If anyone can help or needs more detail about that issue I'll happily go into more detail.
If I understand your problem correctly, it sounds like you may have soft select enabled. Try hitting the "b" key and moving your points again. Or double click on the move tool and look to see if Soft Select is enabled. Depending on the falloff, you might see a colored gradient across your model when you select a point.
I did try that prior to making this thread, but you're on the right track.
It's almost like Soft select is enabled but there is no colour gradient, and in the Move Tool Options all the Soft select boxes are unchecked, so it has nothing to do with that as far as I'm aware, all i remember is missing a few buttons on my keyboard and all of a sudden it turned into a Soft select, but without the options checked.
Hmm, you don't have Smooth Mesh Preview on by chance? The "1" key will toggle back to the poly cage if so. 2 will show the cage and smooth mesh, and 3 will just show the smooth model. Moving a point on a smoothed polygon will give a similar effect. Also make sure you haven't inadvertently selected a NURBS object. Just trying to eliminate possibilities. In the mean time I'll keep digging...