I agree that when a bag is checked at the airport, and the bag is tagged, that in your airline reservation (whether viewing it on the web or a mobile device) the record should list for the pax the bag tag# which is printed on the claim check. There should also be a link in either a web or mobile app view for live tracking of the bag. Actually now both US and DL have this live internet tracking of bags so you can see e.g. when it was loaded on to the plane. With US you can retrieve the bag tag lookup via your 6 character US PNR. So if you did not readily have your claim check you could track the baggage using the confirmation# which you likely do have on your boarding pass.

I actually came across this thread as I was doing a google search for an iOS app that could scan an airline bag tag claim check and convert the barcode in to the bag tag#. Does anyone know if there is a barcode scanning app that can read the airline bag tag barcodes?

Also what I personally do is when I check a bag I will add the bag tag# to the flight notes on TripIt:
http://tripit.com
You can add these notes from the TripIt iOS app, and then they are in sync so anyone authorized to view your trip either on the web or another mobile device would be able to view the notes that you enter. But I was jjust looking for a more automated work flow than to have to manually type the bag tag claim# or #'s in to Tripit from the iPhone.