A design on laptop drives has the spindle fixed vertically in the center of the tray, instead of having it move up into the hole on the disc. The disc snaps on to the spindle, just as it would snap into a DVD case, keeping it from falling. These are usually in small form factor desktop PC's also.
Another design used in most desktop PC's is to have 4 small tabs on the tray that hold the disc vertically until the tray is closed.