I have always stored my console vertically - that is, with the disc tray perpendicular to the ground. However, I was wondering if it's safe to store the console horizontally, where the disc tray would be parallel to the ground?
I've always stored mine vertically, but that's just to save space on the shelf it's sitting on. I think the Xbox 360 console is about like the Playstation 2 and 3 consoles, which is to say that it can be stored either way without damage. Like Sollyhol said, just don't move it while's it's on and running.