I think it really depends on the Music Ministry structure.
I my case, the M.O.M. is over all in charge of each choir (and praise & worship team). There is a president who handles the administrative aspect of the choir, (i.e. outings, rehersal changes, corrospondence), behavior issues, etc. and communicates with the M.O.M.
However, we also have several directors who are gifted to teach and direct. They have no "formal" task but only to direct and teach songs of their choice (and the Lords), but the song has to go through the M.O.M. first, if it is not scripture based and doctrinally sound, the M.O.M. is NOT having it.
As for who picks songs for Sunday services, the M.O.M. works closely with the director(s) of each choir and together they come up with Sunday service selcections, but M.O.M. has the final say so on the selections.