Vertical (straight up) and fixed to a mast or other stable mount, as high and as far away from computers or florescent lighting as possible. Why? Because the people who build these things and know more about radio waves than I do all mount them this way.
When you think of it, though, radio waves peak and trough more or less perpendicular to a level plane so you would want the maximum surface area of the antenna meeting the wave (at least that is what I recall from two semesters of college physics - but this was back in the early 80s).
As far as in what direction, I was instructed here on this site to aim my j-pole antenna so that the lower arm is closest to the transmission towers of the frequency I want to receive, like so (cue cool ascii art) :
| | <------ source of waves (broadcasting antenna/airport)