When I wrote my master's thesis, a professor who read it said that I should not use the phrase "A function of class $k$." but instead "A function of class $C^k$". I am not an expert about mathematical history of notations, but I read that in Geometric Measure Theory, H. Federer actually uses the first one, and it seems logical for me: I think that $C^k$ is the abbreviation for "of class $k$". Therefore, employing "class $C^k$" seems like a repetition. Or maybe the other notation is just not used any more and should simply be prohibited?
Federer was not exactly known, even to his contemporaries, for employing standard notation. Here is a quote from Steenrod's 1948 Math Review of some mimeographed notes of Federer for a course on differential geometry.
The most striking feature of the book to the casual reader is the notation. The author adopts the view that certain familiar notations are misleading, and obscure the meanings of definitions and theorems. He replaces them by more elaborate notations based on the roots in set theory of the concepts represented (e.g., the polynomial x becomes the sequence of its coefficients [0,1]). A few such changes would not be worthy of comment; but he has carried out the prodigious task of applying the same stern standards to every phase of the work. The result can be described by saying that a resemblance to any notation, living or dead, is purely coincidental.