Well in my case it's definitely the bass picking up something.. when I turn down the volume on the bass there is no noise. The Suhr and the Fender Victor Bailey act about the same, which makes me wonder, since they are both designed by John Suhr. With other basses I don't have this problem...
I can't offer much advice. I don't have issues, but with this specific band I'm currently using EMG (and Spector), which if you're going for true singles probably aren't up your alley.
My Suhr is more or less a straight up J bass with passive single coils and I haven't found it to have any more or less hum than any other J bass.