I was thinking about this when doing my DBS check recently. I have a passport and driving licence, but they also want your NI number.
I also discovered I don't know where my birth certificate is (though I know I could order another one). My council tax bill would do for proof of address at the moment, as it's less than three months old. I think our latest water bill is probably less than three months old too, but we only get those twice a year.
Other than that I would be struggling for proof of address. It really is time we had a standard ID card that would be universally accepted.