The answer changes drastically between 1800 and 1900. In the early-to-mid part of the century it was still believed that bathing could cause illness and access to clean water was limited. Those with servants often had tubs, along with hotels, and would have water brought in by hand (which never stayed warm long) and even then more than one person would use the same water. Mostly, bathing consisted of a pitcher of water and a washcloth applied to exposed areas, on a semi-regular basis.
By the end of the century the advent of plumbing and more advanced medical theories motivated greater cleanliness in the populace. However, even then, the poorer classes would not bathe nearly as often as the more well-to-do, perhaps only in warm weather because they had to make do with outdoor sources of water or wells.
In the "wild west" bathing was done in streams, ponds, or lakes, but mostly only for men. Women would still have been largely constricted to indoor bathing options (pitcher of water) and maybe a rare dip in the stream (for special occasions, such as a wedding). And again, as the century progressed, this would have changed along with technology and medical science. By 1900 most people knew they should bathe more often/frequently even if they didn't have the means to accomplish it.
My Mother told me that the farm where her grandparents lived, in Ohio, her grandfather would take a bath every Spring, when the weather first got warm. All winter he kept on his long johns, with the buttons on the back. I always assumed he bathed fairly regularly after that, maybe Saturday night in preparation for church the next day. When you add to this the fact that toilet paper was not invented until late in the 20th century- LOL- it really adds to your enjoyment of the olden times, huh? I bet the cowboys would pay for a bath when they got in from the range, but the trip from Texas to Kansas would take over a month. My guess would be at least several times a year, for most folks. But certainly much less than we are accustomed to. When I toured the castle of Versailles outside Paris I was shocked that there was not one bathroom in the whole palace. No toilet, no tub. If royalty was not bathing in France, people in the wild west were not bathing much.
not often. they wore tight clothing, tight collars, they kept away from sewage, they changed their clothes anbd laundered them, and no one squirted soda on them.
In the blood of kittens.
Cowboys out on a cattle drive would normally find a stream or river to bathe in if the opportunity presented itself, using bars of homemade soap kept in their packs. However you need to understand that people back then really didn't have the standards of hygiene we do today. Some would go without bathing for weeks at a time. Crotch Rot was common, as were a plethora of other dermatological conditions from lack of proper hygiene. It was widely believed that if you bathed too much, you could get sick from it.
At home or in town, water was usually heated over a fire and poured into a tub, and/or the tub itself was suspended over hot coals, or one would find a nearby water source to bathe in.
Not often enough, maybe once every two weeks?
I hate your generalizing question