Jos. Chas. AINSWORTH's Chicago voter registration and child's brth certfiicate both list his borth place as Wyoming. He would have been born around 1856. I understand that the population of WY at the time was very low, and that it was part of Nebraska Terr. Can anyone tell me where to look in the 1860 census? Other databses?

You are correct that Wyoming was included underneath the canopy of Nebraska for the purposes of the 1860 census. Wyoming residents (10 pages worth) are accounted for at Fort Laramie Reservation (when checking the census just for Nebraska you select "unaccounted" counties to find Fort Laramie. A perusal of those ten pages though does not seem to me to show anyone with the Ainsworth surname. Perhaps Joseph was indeed born in Wyoming but his parents may have relocated elsewhere by the time of the census. The other possibility (perhaps remote) is that he was born in another place called Wyoming in another part of the USA as there are various counties called Wyoming (like in NY state).