Gas regulator issue, or Fridge?

I doubt if it's your fridge. Generally speaking the more use/flow thru a reg. will cause more singing(whine). I would think that your should be able to duplicate the sound using an alternate source of flow as rdhetrick mentioned. Though I've not noticed my reg singing, I do know factually that all natural gas regs sing under use. I may not hear mine 'cause I'm somewhat hearing impair.......well, my wife thinks so anyhow. You can have your pressure checked, or check it yourself for lock-up(no flow) and for flow. Should be real close to 11" water column when flowing. Or you can change it and see/listen if the new one sings to ya.