The "facts on the ground" do not force us to use one expression or another. What we say expresses our view of the facts.
Both forms of the present perfect make a statement about the world up until now.
You could use have been singing:
I have been singing in French ever since I began taking singing lessons.
to emphasize that at no time did you abandon that practice since first taking it up. It began back then and has continued.
Few native speakers would take that to mean that you have never stopped singing, not even for meals or sleep, but a comedian like Groucho Marx might ask "So how do you sleep?"
You can also use have sung
I have sung in French ever since I began taking singing lessons.
to indicate that on those occasions when you sang, it was in French, and this has been true since you began taking lessons. Groucho would not be able to make the same quip there.