What rate of twist would you look for in a 308 with a 22" barrel.
It's offered in 1:10 , 1:11 , 1:12 depending on the manufacturer ?

A slower rate of twist will work better with shorter(lighter) bullets. Typically a .308 would have a 1:12 twist. But I've heard of more rifles with the same twist as a 30-06, 1:10, to be accurate with a wider variety of bullet weights.

From Shilen's website:
.308
- 8" for bullets heavier than 220gr.
- 10" for bullets up to 220gr.
- 12" for bullets up to 170gr.
- 13"* Ratchet rifled 4 groove
- 14"* for bullets up to 168gr.
- 15"* for bullets up to 150gr.
- 17"* for bullets up to 125 gr.

No velocities are given by Shilen, Pac-Nor or Hart. But it's safe to say that if your 308win is 1:12 and it can't can't push a 220gr bullet fast enough to stabilize it unlike the magnums. But, IMO if your trying to shoot that heavy of a bullet you may want to get a magnum. I must admit that i like the 1:10 twist for it's wider range of bullet weight choices, unless it's for a single purpose rifle. But think it's often being forgotten the faster twist's lose the ablility to shoot the lighter varmint bullets. Which my short mag is 1:12 and shoots 110gr V-max @ 3775fps (chrony'd) almost as well as my 165gr match. But if were 1:10 would it shoot the 110's as good, idk.

Okay, I'm not sure where a question about a .308 has introduced the possibility of a "magnum"...

If you're shooting a .308, it depends upon your intended use. If the heaviest bullet you intend to send downrange is 150 grains, a 1:12 would be fine. If you're sending something 175-178 grains and heavier, you need to lean toward 1:10. I know a lot of guys who are shooting .308s with 1:11 (and fractions) as a compromise, but I don't see that they work quite as well as a 1:12 with light bullets, or a 1:10 with heavy bullets.