What could cause large discrepancies in reported data use? On my Galaxy s3, 3 different reports on (cellular) data usage are giving me 3 wildly different answers.

My data counter widget reports 1.26GB transferred, up and down total. 
Android's internal data usage screen says 1.46GB transferred.
And my carrier (via their app) says 1.332GB transferred.

What would cause these to disagree to this extent?
your data counter widget most likely summed-up the exact amount of data transferred. i.e. for a transfer of 33 kB, and another of 17 kB, it would count 33+17 = 50 kB
your carrier more likely sums up blocks of 10 kB. So given the same example, he calculates 40+20 = 60 kB
not for real, but it looks like Android's internal takes both and makes an average, so for our example it would come up with 55 kB. Makes absolutely no sense (and I doubt it's true), but rawly matches the numbers given in your question...