At what point had all available land in the United States been assigned to specific individuals or groups? That is, a legal document (e.g deeds, court rulings, things such as Indian reservations etc.) saying that a specific area of land belonged to Person X?
You are referring to the Homestead Act of 1862. Here is a nice article on the last homesteader, who claimed land in Alaska between 1974 and 1988 (it took a while to finalize the paperwork).
All available land isn't assigned to individuals or groups, the US government still owns a significant portion of the total available land, for example almost all the land in Alaska. One way ho to get "free" land from the government were the Homestead acts, the first of which was passed in 1862. Homesteading was ended by the Federal Land Policy and Management Act of 1976, but continued until 1986 in Alaska. The last claim under the act was made by Ken Deardorff, who filled the claim in 1979, but did not receive his deed until May 1988.
There was a several acre "gore" of land claimed in New Hampshire in the fall of 1983. It had been unclaimed by any town due to survey error. One of the landowners adjacent to the strip discovered the error. bought the land on the other side, and then filed for a title amendment.