I built a pole barn 8 years ago and screwed through the ridges on the roof (flats for the siding). It held up great in a near miss with a tornado (I lost a bunch of 6 year old shingles off the house in that storm). There have been no leaks and no problems with the pole barn roof so I did it the same way (through the ribs) when I roofed my house and garage a few months ago.
As far as I can tell, the only downside with going through the ribs is that it is much more challenging and slower to install and keep it square on the roof. If you put too much pressure on the screw, it will spread the rib slightly. If one end of the sheet's ribs get deformed more than the other, then the sheets will get out of square with the roof. Some websites say that you can't get a good seal on the rib, but as long as you center the screw perfectly on the rib, it will seal nicely.
Another negative is that it is harder to walk on safely (screwing through the flats will give you a good foothold -stepping in the screw heads).
The upside of going through the ribs is that the potential for leaks is greatly lessened. Metal sheets expand and contract daily as they heat and cool- this can actually elongate the screw holes over the years, which can eventually lead to a poor seal. A poor seal is more likely to leak when exposed to more water (like if the screw is on the flat). A little water leaked during every rain over the years will eventually rot out the wood around the screw and it will no longer hold.
I know that 95% of the metal roofs in my area are screwed through the flat- it is definitely the quickest and easiest way to install the roof.
Most manufacturers do say to put screw it through the flat but interestingly enough they say to nail it through the rib.
Screwing it through the flat is easy, fast and looks good (you can't tell when from looking at it from the ground if the screws aren't perfectly aligned like you can with screws in the ribs). My concern was more about long term durability (50 years down the road) and that's why I screwed it through the ribs.