Shall

Becoming old-fashioned) used with “I and We” for talking about or predicting the future: This time next week I shall be in Scotland. We shan’t be gone long. I said that I should be pleased to help.
Used in questions with “I and we” for making offers or suggestions or asking advice: Shall I send you the book? What shall we do this weekend? Let’s look at it again, shall we?
(Old-fashioned or formal) used to show that you are determined, or to give an order or instruction: He is determined that you shall succeed. Candidates shall remain in their seats until all the papers have been collected.
Shall / will
In modern English the traditional difference between “Shall and Will” have almost disappeared, and shall is not used very much at all, especially in NAmE. Shall is now only used with “I and We”, and often sounds formal and old-fashioned. People are more likely to say: “I’ll (= I will) be late and you’ll (= you will) apologize immediately. No I won’t!”
In BrE shall is still used with “I and We” in questions or when you want to make a suggestion or an offer: What shall I wear to the party? Shall we order some coffee? I’ll drive, shall I?

Should

Used to show what is right, appropriate, etc., especially when criticizing sb’s actions: You shouldn’t drink and drive. He should have been more careful. A present for me? You shouldn’t have! (= Used to thank sb politely)
Used for giving or asking for advice: You should stop worrying about it should I call him and apologize? I should wait a little longer, if I were you.
Used to say that you expect sth is true or will happen: We should arrive before dark. I should have finished the book by Friday. The roads should be less crowded today.
Used to say that sth that was expected has not happened: It should be snowing now, according to the weather forecast. The bus should have arrived ten minutes ago.
(BrE, formal) used after I or we instead of would for describing what you would do if sth else happened first: If I were asked to work on Sundays, I should resign.
(Formal) used to refer to a possible event or situation: If you should change your mind, do let me know. In case you should need any help, here’s my number. Should anyone call (= if anyone calls), please tell him or her I’m busy.
Used, as the past form of shall when reporting what sb has said: He asked me what time he should come. (= His words were: “What time shall I come?”) (BrE, formal) I said (that) I should be glad to help.
(BrE) used after that when sth is suggested or arranged: She recommended that I should take some time off. In order that training should be effective it must be planned systematically.
Help Note: In both NAmE and BrE this idea can be expressed without “should”: She recommended that I take some time off. In order that training be effective…
Used after that after many adjectives that describe feelings: I’m anxious that we should allow plenty of time. I find it astonishing that he should be so rude to you.
(BrE, formal) used with I and we in polite requests: I should like to call my lawyer. We should be grateful for your help.
Used with “I and we” to give opinions that you are not certain about: I should imagine it would take about three hours. “Is this enough food for everyone? I should think so.” “Will it matter? I shouldn’t think so.”
Used for expressing strong agreement: “I know it’s expensive but it will last for years. I should hope so too!” “Nobody will oppose it. I should think not!”
Why, how, who, what + sb/sth do use to refuse sth or to show that you are annoyed at a request; used to express surprise about an event or a situation: Why should I help him? He’s never done anything for me. How should I know where you’ve left your bag?
Used to tell sb that sth would amuse or surprise them if they saw or experienced it: You should have seen her face when she found out!
Should / ought / had better
Should and ought to: are both used to say that something is the best thing or the right thing to do, but should is much more common: You should take the baby to the doctor’s. I ought to give up smoking. In questions, should is usually used instead of ought to: Should we call the doctor?
Had better can also be used to say what is the best thing to do in a situation that is happening now: We’d better hurry or we’ll miss the train.
You form the past by using should have or ought to have: She should have asked for some help. You ought to have been more careful.
The forms “should not or shouldn’t” (and ought not to or oughtn’t to, which are rare in NAmE and formal in BrE) are used to say that something is a bad idea or the wrong thing to do: You shouldn’t drive so fast.
The forms “should not have or shouldn’t have” and, much less frequently, “ought not to have or oughtn’t to have” are used to talk about the past: I’m sorry; I shouldn’t have lost my temper.
Should / would
In modern English, the traditional difference between “should and would” in reported sentences, conditions, requests, etc. has disappeared and should is not used very much at all. In spoken English the short form: “‘d” is usually used: I said I’d (I would) be late. He’d (he would) have liked to be an actor. I’d (I would) really prefer tea.
The main use of should now is to tell somebody what he or she ought to do, to give advice, or to add emphasis: We should really go and visit him or her soon. You should have seen it!