My Moss Reid character is a private investigator, a private eye. The profession may be as old as the hills, but how old is the label / title?
A quick search in the Google Books Ngram Viewer shows that “sleuth” and “private detective” were far more widely used terms during the late Victorian age. They continue to be very popular, at least within the world of books.
The rise of “private eye” and “private investigator” as terms could be more of a post-war thing, taking off in the late 1940s.
The single-word term “detective” has far more hits than any of these terms, but would obviously span both police detectives and PIs in the private sector.
As for “shamus”, I don’t think it’s used much within Ireland, probably because it would be confused with the name Séamus. It gets a much lower score in the Ngram Viewer, as does “gumshoe”.