What is the origin of scut in scut work?
According to Merriam-Webster,
scut work: routine and often menial labor
Probably from medical argot, scut meaning 'junior intern'
First known use: circa 1962
... women who generally feel that they are second-class citizens in the medical profession -- unappreciated and directed by (mostly male) doctors to perform largely scut work.
This ngram shows that scut work was not used before 1960 when it took hold as medical jargon. It supports the idea that the term was not already in general use and subsequently popularized in hospitals, rather that it originated there.
This returns us to the medical coinage. Could it be related to meaning (2) in the OED?
1. the short tail of a hare, rabbit, or deer.
2. (inf. chiefly Irish) a person perceived as foolish, contemptible, or objectionable.
There is another theory that it is derived from Oxford University servants called scouts, but I consider this unlikely given the US origin of scut work.