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Let's say I want to arrange a lesson with my coach and I say:

Shall we have a lesson on Monday?

I understand the use of 'shall' in American English is considered to be formal, whereas this is not the case in British English.

I would like to understand the different connotations (is it formal? is it a request for confirmation? is it a suggestion?) that the following questions have:

A. Shall we have a lesson on Monday?

B. Can we have a lesson on Monday?

C. Should we have a lesson on Monday?

D. Could we have a lesson on Monday?

E. Are we having a lesson on Monday?

F. Will we have a lesson on Monday?

G. Are we going to have a lesson on Monday?

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There are some shades of meaning in the questions you've listed. But before I go through them: the most important thing in interpreting any of them will be context.

That said, I think the most useful way of splitting these up would be to say in what context you'd be most likely to hear them.

A. Shall we have a lesson on Monday?

As you correctly point out, you will probably only hear this in UK English, where it is a polite suggestion. An American equivalent might be:

How about we have a lesson in Monday?

or, slightly more aggressively but still polite:

Let's have a lesson on Monday!

Your "B" and "D" sentences will also probably be heard as a suggestion:

B. Can we have a lesson on Monday?

D. Could we have a lesson on Monday?

Here there's an implication that you're consulting the other person's schedule. You might hear:

Your next lesson is scheduled for Monday, but I have a dentist appointment, so I'm afraid I can't make it.

Well, then, can we have a lesson on Tuesday instead?

"Can" can also be used to ask if something is possible:

The Vice President will be visiting this week, and the whole campus is on lockdown, including the rehearsal rooms.

Oh, no! Can we still have a lesson on Monday, then?

while "could" implies a stronger desire (in American English, at least--in British English I don't think that's necessarily the case).

How would you like a lesson from the great Zanzini?

Ooh! Could you give me a lesson on Monday? I'd love that!

"Should" is interpreted in its usual meaning of "is it advisable to"/"is it a good idea to".

You need a lot more practice before the audition on Tuesday.

Well, then, should we have a lesson on Monday? Or should I rest my voice?

The last three:

E. Are we having a lesson on Monday?

F. Will we have a lesson on Monday?

G. Are we going to have a lesson on Monday?

are simple questions about whether an event will happen; you would probably hear them used in a case where there are a series of regularly scheduled lessons and you want to confirm that the next in the series is going to occur. For example:

This class will meet every Thursday from now until the end of December.

Are we going to have a lesson on November 27? That's Thanksgiving day.

or

I'm going out of town for the weekend, so I won't be able to grade your paper until at least Tuesday.

Okay. Are we still having a lesson on Monday?

or

I want you to spend all of your time between now and Thursday practicing your forehand.

Okay. Are we having a lesson on Monday, or should I just work on it on my own?

Any difference in which one of the last three you're more likely to hear will have more to do with regional variations than shades of meaning.

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    This is mostly perfect. I would propose B and D are not suggestions, they are requests; the suggestion is "Let's have a lesson on Monday!"
    – Codeswitcher
    May 5, 2014 at 18:52
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    In American English, at least, if you know you are going to have the lesson sometime, I would consider "Can we have it on Monday?" to be a suggestion--I don't see any real semantic difference between that and "How about Monday?".
    – chapka
    May 5, 2014 at 21:37
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    You don't? I certainly do. While "Can we...." and "How about..." both indicate the speaker's preference, the fact that "Can we..." is, strictly speaking a yes-no, boolean question, makes it much stronger, moving from the mere, "I would prefer..." to "Will you accede?"
    – Codeswitcher
    May 5, 2014 at 22:36
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    Then there are "May we have a lesson on Monday?" and "Might we have a lesson on Monday?"
    – BobRodes
    May 6, 2014 at 13:11
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    Why do you say you don't hear "Shall we" in the U.S.? Look at this Ngram. On the other hand, I think first person questions are the main place "shall" is still used in the U.S., and only in a fairly formal register.
    – Peter Shor
    May 7, 2014 at 17:59
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From a British English perspective:

A. Shall we have a lesson on Monday?

Formal, and a little archaic. Asking if there is going to be a lesson on Monday. It might be the teacher, asking whether one is desired, or the pupils asking if it will be held.

B. Can we have a lesson on Monday?

Formally, this asks whether it is possible to have a lesson. Informally it is likely to be used (misused?) to request one.

C. Should we have a lesson on Monday?

This is asking whether or not having the lesson would be a good idea.

D. Could we have a lesson on Monday?

Almost the same as B. Strictly it's conditional, so the condition should be stated: "Could be have a lesson on Monday if we cancel Wednesday's?" In practice, though, that subtlety gets forgotten and it's pretty much interchangeable with B.

E. Are we having a lesson on Monday?

F. Will we have a lesson on Monday?

G. Are we going to have a lesson on Monday?

To all intents and purposes, completely interchangeable. The difference from A is that in the case of A it might not yet be decided; these assume that it is already decided, the person just wants to know what the decision is.

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  • Can "shall" be used in a question in an informal context? For example, to invite a friend to go to the cinema: "Shall we go to the cinema?" Or this still sounds formal and a little archaic.
    – Nico
    May 7, 2014 at 7:01
  • What would you suggest is the most common way to arrange a lesson in a colloquial context?
    – Nico
    May 7, 2014 at 7:02
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    I tend to be formal and a little archaic, so I might well say "shall we go to the cinema" :) If I pushed myself to be informal I might ask, "Do you fancy going to the cinema tonight?" or, "How about going to the cinema tonight".
    – digitig
    May 9, 2014 at 16:56
  • Just to confuse matters slightly, in Scottish English (a subtype of British English), you'll almost never hear "shall" and "will" takes it's place, so everything above looks right to me with the exception of "will we have...", which for me would be grouped in with "shall we have...". Aren't dialects fun? ;)
    – Alan Third
    May 10, 2014 at 23:46
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I'm not sure how to answer this by narrowing it. Is this question broad? Anyway...

I'm neither BrE speaker nor am I an AmE speaker. What I see these sentences is through InE which follows BrE. So, I think I may contribute here.

IMO...

Shall we have a lesson on Monday? - Asking with a bit of demand or assertion
Can we have a lesson on Monday? - Asking in general (also asking with a pinch of probability).
Should we have a lesson on Monday? - Expresses some logical, emotional, practical (or something the like) way of learning that lesson as in "You should have called me.*
Could we have a lesson on Monday? - this is clear. Asking with politeness.
Are we having a lesson on Monday? - Asking with a bit of affirmation.
Will we have a lesson on Monday? - Seems bit weird to me. But still, it's requesting an action.
Are we going to have a lesson on Monday? - It seems reminding someone that is already decided as in "Are we going now (finally)?*

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    For AmE, "Are we having...", "Will we have...", and "Are we going to have..." are not requests to arrange a lesson; they are inquiring as to whether a lesson has already been arranged.
    – Peter Shor
    May 3, 2014 at 17:06
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    In my experience, in BrE, "shall" can be used as a suggestion, e.g. "Shall we go to the cinema tonight?".
    – Nico
    May 5, 2014 at 11:40
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    Also in my experience, in BrE, as @PeterShor describes in his comment, "Are we having...", "Will we have...", and "Are we going to have..." are actually asking for confirmation. I imagine there may be a subtle difference in use between them and it would be great if someone could explain what the difference is.
    – Nico
    May 5, 2014 at 11:44
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    @MaulikV: I find Peter Shor's comment to apply equally to BrE. However, I don't find that the distinction is as clear-cut as he suggests. "Shall we have pancakes?", is asking whether we want to have pancakes or not, in both BrE and AmE. "Are we having/going to have pancakes?" can be (and probably is) asking the same thing. So it depends on context: "Are we having/going to have a meeting on Monday?" would very probably be a request for confirmation as Peter has it. Again, though, these don't vary between AmE and BrE in my experience.
    – BobRodes
    May 6, 2014 at 14:09
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    @Nico added sources wherever they were possible.
    – Maulik V
    May 9, 2014 at 16:16
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This is a good summary of the uses of shall and will in English. This little excerpt is interesting:

An illustration of the supposed contrast between shall and will (when the prescriptive rule is adhered to) appeared in the 19th century, and has been repeated in the 20th century and in the 21st:

  • I shall drown; no one will save me! (expresses the expectation of drowning, simple expression of future occurrence)
  • I will drown; no one shall save me! (expresses suicidal intent: first-person will for desire, third-person shall for "command")

I note (and so does the article) that the "prescriptive rule" is often not adhered to, as other posters have elaborated on in detail.

I also remember reading something from Alfred the Great that began "Deos boc sceal to Wiogora Ceastre", meaning that this book is to be delivered to Worcester. This is a (very) early use of "shall" to mean a requirement, as in "The size of the element shall not exceed 640x480 pixels".

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"Shall" is used in questions for offering to do something for another person. E.g.,

Shall I carry that heavy case for you?

It is also used in questions for suggesting possible actions and asking if an idea is good. E.g.,

Shall we stop now?

What shall I do?

Shall we tell her?

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  • Would you say all those uses you list are formal? Could you imagine a context where "shall" is used in a question in an informal context?
    – Nico
    May 7, 2014 at 6:48
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    No. I didn't say such sentences using "shall" are formal. Compare those to the following ones: I "shall be" in touch soon. Or, I "should be" grateful for some help. These are examples of formal sentences. In the first one "shall" can be replaced with "will" and in the second one "would be" can be used. This is what has happened in modern English.
    – Mohammad Nazar
    May 8, 2014 at 7:56

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