We all know what shelters are NOT historically correct, so let's get into which ones were.
The easiest is the tipi. Of course no mountain man would have ever drug one of these shelters along with him while he trapped, but if he had married into, or had become accepted into, a tribe, it was certainly possible that he would have spent some of his time in a tipi.
I don't want to go too deep into this issue, because while the tipi itself is correct for the fur trade period, historical reenactment is about the details. Painted tipis were the possession of their owners; they would never have been available to a fur trapper.
Also, take a good look at the possessions inside a tipi. Beds were made of pairs of backrests and buffalo robes. Clothing was stored inside rawhide parafleche's (not boxes - which weren't around until the late 1860's), and items hung inside the lodge were kept to a minimum. Remeber, these Natives people were nomadic - they were always prepared to move at a moments notice. Wealth was dependent upon horse ownership and what they would carry. The typical fur trapper had perhaps two or three horses, which were often overburdened with traps and pelts. Trappers were a "poor" lot of folks to the Natives.
So, where did the mountain men go for shelter? Most likely a cloth tarp or buffalo robe tied above them. Again, these men were after beaver hides. They made camp only long enough to collect the beaver they had caught, skin them and stetch them. Then they moved. Camp was a matter of necessity, not convience or luxury.
Since the fur trappers were considered "poor" by the Native people, it is possible that the commonly referred to half-dome shelter originated with a tribal member allowing a mountain man to use their "sweat lodge" as a form of shelter. Easy enough. Throw some blankets or hides over the frame, and you've got a nice shelter from the elements.
Personally, this is the shelter that I have used for years. I have shaped my own frame-poles that I travel with and reuse i the interest of preserving wild areas, but it is quite easy to see how cutting fairly small willow bends and tying them together to form a frame would have taken the least amount of work and effort. Then throwing some blankets or hides over the top and anyone has got shelter from the wind, rain, and even snow. In modern times, I use the three canvas linings from a 18' tipi to form the cover of my shelter. They can be arranged in a variety of ways, from producing 100% privacy and shelter, to allowing half of the dome to be open to cool summer breezes or a nice warm fire.
A trick that I use toi stay warm at night is something I'm sure many people have used before. Once my dinner fire has died down to cools, I'll place a few round rocks the size of two fists into them. After about an hour, when sleep is calling loudly, I'll transfer these rocks into the dome shelter where they keep the night cold at bay well into the morning.
The canvas folds up into a bundle that is about 18"x18"x 12" and weights about twenty pounds. As long as I have the willow branches, the only other item I need is a ball of twine or rope. This shelter has withstood tornado-strong winds (roes thrown over the top and staked down), as well as winter storms that drop nearly a foot of snow. I have slept six men inside the shelter in a pinch, but routinely only have myself, wife and daughter. There is plenty of room. The frame work provides a place to hang a few items, the canvas allows the interior to remain bright during daylight hours, and with a door, we have all the protection and privacy we need.
The interior floor is covered with three buffalo hides, all well used. And we sleep under a number of wool blankets (not a Whitney 4-point).
