What have you found to be the most effective method of editing music in this job? For instance, in what order do you work, and what types of mistakes do you look for?

While we will all have our own approach, I tend to explain that we need to edit both horizontally and vertically, looking not only for mistakes in the current score (as compared to the principal sources or the engravers score), but also for any notation that doesnt make sense, or is questionable, in its context. The dimensions of the score to carefully check are: pitch, rhythm, dynamics, articulations, clefs and key signatures, tempo and meter, pedal and percussion markings, lyrics, uniformity between parts, rehearsal and measure numbers, text (all other), extra (specific to the work), typesetting (overall look of the page). Please feel free to use your noggin if something looks odd.