Thank you for the quote (and the compliment), Chris. Much appreciated. Well, after a great deal of consideration, I've decided to agree with myself ;) ....... my advice to learners is "Don't try to use 'shall' ".
In spite of what many native speakers believe, 'shall' is not archaic. It's not rare, it's not literary, it's not pretentious, and it's not even formal. Lots of people - mainly outside the US and Canada - do use it in their daily lives, even in informal situations.
The problem is that very few people know the rules about how to use 'shall'. North Americans and people who've been taught by N. Americans will tell you that it's used in formal instructions or when you're 'goofing around' (pretending to be sophisticated a for a laugh). Meanwhile, teachers and students from countries which have outdated textbooks (particularly Russia) will quote rules and paradigms which fell out of use in the 1960s. The result is that most people who try to use 'shall' end up using it inappropriately.
Many native speakers will also tell you that we "always use 'will instead of 'shall' these days". This is also untrue, because it doesn't work with questions. Take a look at the examples which Chris quoted.
As Chris pointed out, a situation in standard English where 'will' is definitely wrong and 'shall' is defininely right is in first-person questions, where the speaker is making a suggestion or an offer. Typical examples are "Shall I carry your bag for you?" or "Shall we have lunch?". You cannot use 'will' in those questions. These are common in British English, and you will find them even in modern textbooks.
However, even in these situations, you don't have to use 'shall', because you could equally well use 'Should'. American English speakers use 'Should we..?", and so do an increasing number of British English speakers. You have to have a 'sh..' word for first-person questions, but it doesn't need to be 'shall'.
Make life easy for yourself. Forget about 'shall'.