I want to be able to hear if a shiny has appeared but prefer my own music collection to that played during game play.
As others have stated, it is not possible to individually adjust Sound Effects & Music volume in Pokemon X/Y. Even the Mono/Stereo option has been removed in this version.
On shiny catching however, as a general rule of thumb, if you're not 100% sure if it's shiny, catch it! It's only going to cost you a Quick Ball/Pokeball to check.