The reviews for the 6950 and 6970 are out. The 6970 performs at par with the GTX 570, while the 6950 has a segment all to itself. Given that the 570 costs less than the 6970, while coming with great stuff from nVidia like PhysX, CUDA, and 3D Vision, I'd say "None of the above" to your choices and get a 570 instead. Reasons:
The 480 is too much of a power hog and produces a lot of heat.
At this point, the 58xx series are retired in favor of the 69xx series, so the 5870 is also out.
The GTX 2xx series are outdated (There isn't any GTX 270, maybe you mean a 275).
The 6870 is OK since it has all of AMD's new technology and architectural advancements built in, but if your budget allows it, Go for the 6970 or 6950. Get 2 of each if your budget and PSU allows it.