Is there any problem using a flight service between Schengen countries (with a single-entry Schengen visa)?
As long as you stay completely in the Schengen region, you don't need to re-enter, which means you don't need a multi-entry visa for a flight from Germany to France. It's similar to a domestic flight, essentially there's no difference in terms of visa issues between flying Frankfurt-Munich and Frankfurt-Paris. Obviously you should make sure the number of days you're allowed to stay in Schengen area is not exceeded, neither the specified dates. For more complicated routes, there might be additional issues you have to consider, like when flying to/from Svalbard or entering Andorra, but this is not part of your question.