Spine, physical weight and degree of straightness. Don't care much about diameter. They're all skinnier than wood. I like a medium weight (for carbon) shaft that I can front load. And I want the shafts to be straight within .003 of an inch. Tapered or not doesn't matter to me. I shoot Easton Axis and Gold Tips because they happen to be good fits for my bows. Most other brands would no doubt work as well.
Accuracy is determined by the shooter, not the arrow. Any arrow well matched to the bow will be accurate, if the shooter is.
The skinnier the shaft and heavier the shaft (particularly if the weight is concentrated in the front of the shaft) the greater the penetration, all else being equal.