This article is from the alt.usage.english FAQ, by Mark Israel firstname.lastname@example.org with numerous contributions by others.
The traditional rules for using these (based on the usage of
educated Southern Englishmen in the 18th and 19th centuries) are
quite intricate, and require some choices ("Should you like to see
London?"; "The doctor thought I should die") that are no longer
idiomatically reasonable. But if you're dead set on learning them,
you can access the relevant section of "The King's English" at
outside England has always been different, although the historical
prevalence of "shall" in the U.S. is sometimes underestimated:
Benjamin Franklin said, "We must all hang together, or assuredly we
shall all hang separately"; and the Shaker hymn "Simple Gifts" has
"To bow and to bend we shan't be ashamed."
The old joke, where the Irishman cries for help: "I will drown
and no one shall save me" and the Englishman mistakes this for a
suicide resolution, is contrived, in that an Irishman would far more
likely say "no one will save me."