NODE/DIMAP show five senses of among; ODE shows only four core senses. A subsense in NODE has been sensibly absorbed by its core in ODE. I have followed the sense numbering in ODE. Amongst is treated as a simple orthographic variant, substitutable in all cases for among, and in any case, not likely to occur in modern US English.
Notes from the spreadsheet:
The senses analyzed occur only in the Sense Analysis Summary for now; I have not broken out a separate spreadsheet. No new senses are added.
Other instances of among and amongst that may deserve idiomatic treatment
Plural reflexive pronouns are likely to occur as the complement for senses 3 and 4. When the among PP is first in a sentence, it is likely to be sense 3, though it could in principle be any of the senses.