This was a weird problem: I plugged my headset into the phone, and recorded, and heard a buzz. This didn’t make sense, but, it turned out to be a grounding problem with a sensitive mic.
Normally, when you hear the 60Hz buzz, it’s a sign of improper grounding. The power hum is getting into the signal.
This time around, I had a problem with the iPhone, which runs on battery. It didn’t make sense.
I eventually figured, by using test recordings, that it was caused by touching the mic or handling the cable near the mic. The 60Hz hum was passed through my body, was picked up by the mic, and then put into the signal path.
The fix was to ground the iPhone. How?
Well, normally the phone is pretty close to grounded, by its metal case. The problem I had was that the phone was in a plastic case. It was insulated from the ground.
The fix was to remove the case, and touch the phone.
I’m not sure why this worked, but it worked.
Similar Problems on the Laptop
I had a similar problem on the laptop, and the fix was to touch the USB port.