Is one tenth of a foot a common unit of measurement?
The tenth of a foot is an American measurement. (Because it uses the foot, which is used in America.)
I interpreted the "American" in the question as meaning what is officially called the "U.S. customary" system of measurement.
The name "U.S. customary" was once needed to distinguish the US customary foot from the imperial (British empire) foot.
Since 1959, the "international foot" has been defined to be exactly 0.3048 meters.
The foot can be further divided using the US customary system into 12 inches. This is the usual division in the system.
In general, parts of an inch are measured using powers of
so we see things like
In engineering and machining (tool and die) it is common to use decimal inches, so
(Machining tolerances are often stated on "tenths" meaning ten-thousandths of an inch.)
Similarly, instead of using the more common feet and inches format, surveying and construction (buildings and landscape) often use feet and decimal parts of a foot.
Instead of 15 ft 6 in (or 15'6"), we would be likely to see 15.5 ft.
(1 ft = 12 inches, so 6 inches = 0.5 feet.)
It is possible to buy an "engineers" ruler or a tape measure that is marked in feet and tenths (and hundredths) of a foot.
(Often in addition to the much more common feet and inches and 1/2, 1/4, /18, and 1/16 inch.)
By the way, until 1959, there was a small difference between the "imperial" (British empire) and the "US" foot.
If you want to read more, here is a link to the Wikipedia article.