Pipe tomahawks were popular trade items, and the natives choices were often based on what they thought of as tradition and personal prestige. If an important chief had a pipe tomahawk, having a pipe tomahawk helped make the individual native feel like an important chief, and they would often put that above utility. It certainly makes more sense, from the utilitarian standpoint to carry a fragile pipe separate from a tool, and not to have a hole drilled the length of the already too thin handle of an axe that was needed for camp chores.
European hunters were much more utilitarian, and would be more likely to carry either a pole axe or a spike axe, depending on whether they wanted a hammering end or a digging end on the side away from the cutting end; or even a double bitted one, so one side could always be kept sharp. Settlers who also hunted would be likely to carry something like a belt axe or shingling hatchet, that was designed for use around the homestead.
The now popular single bit throwing tomahawk style with no pole, spike, or pipe end, would have been the cheapest hatchet to make at the time, as it would have taken less iron and less work. They are also a bit lighter to carry. They are also about worthless for driving a tent or snare peg, and digging with one would make it useless for cutting without a good sharpening.
As far as what type was the most common, I think there were a lot of choices and people chose the one that felt right to them. They were all common types.
