A space documentary I was watching mentioned that the last solo flight by an American into “space” (in this case, “space” is defined as at least 100 km/62,5 miles above the Earth’s surface) was that of Gordon Cooper in a Mercury flight, in May 1963. Some Soviets flew solo at later dates.
Was this statement correct?
I was thinking more either of NASA (or the equivalent)-controlled missions, or being in a craft which made (or was capable of making) at least one orbit. (of course - there were Apollo astronauts circling the moon solo, but they were launched with partners).
Technically it would have to be one of the latter X-15 flights (lsst in 1967) assuming they hit the 60 mile limit.
Shenzhou 5 was China’s first manned space flight (launched October 15 2003) and it carried a single crewman (later flights have had crews of two or three).
The six Mercury flights were the only US, solo orbital flights.
The widely used Kármán line definition used by Fédération Aéronautique Internationale (FAI) and the US Federal Aviation Administration is 100 km or 62 statute miles (54 nautical miles); only three X-15 flights exceeded that altitude, all before 1964 but two occurring after the end of the Mercury program The 80 km limit was used to justify giving “astronaut wings” to other X-15 pilots but is not universally accepted. @pkbites is correct that Melvill is the last solo American citizen in space at an apogee of just over 102 km.
125 km circular altitude is the lowest altitude where an orbit could be maintained for any duration, and the only solo flights to reach that altitude were Vostok and Project Vostok flights (not counting solo Lunar orbits during the Apollo Lunar missions, of course). Vostok 6, the first flight of a woman crew (cosmonaut Valentina Tereshkova) was in 16-19 June 1963, after the last Mercury flight (callsign Faith 7, 15-16 May 1963), so Vostok 6 would be the last solo orbital flight, but the o.p. is correct that Cooper was the last solo American astronaut in orbital space.
[edit] I forgot about the Chinese space program. You are correct that Shenzhou 5 was the last solo orbital mission. 14 orbits isn’t much by modern standards but it is actually more than all but the last Mercury missions combined.
Stranger
Nitpick: the first two manned Mercury flights (Shepard and Grissom) were sub-orbital.
D’oh! You are correct, of course. 
Did any of the supply flights to Mir or the ISS have a single occupant, either on the way up, or the way down?
Nope. The last Russian solo launch was a joint mission in 1969. Soyuz 4 launched with one cosmonaut and Soyuz 5 launched with three. The two spacecraft rendezvoused in orbit and two men transferred from one vehicle to the other. They returned to Earth that way.
Many Soyuz spacecraft have visited the Salyut, Mir and ISS stations, some with a crew of two, some with three. Some of the two person missions may have carried a small amount of equipment or supplies in place of the third cosmonaut. I don’t know for sure, but I would think they wouldn’t want that extra bit of carrying capacity to go to waste.