I have a Canadian passport, the expired one contains my residence visa for the U.S. UNCIS was very clear in their instructions to me that when I travel abroad I need to carry both passports (new and expired) for re-admission to the US. When my visa renews (in right now), the new I-94 will go in my current passport and only then can I no longer carry my expired visa.

The couple of times I have travelled internationally with it since I got my new one, only at LAX did they want to see the old visa and passport - the other time they said it was ok, they didn't need to actually see it.