1. Can
1.1. Ability or possibility
1.1.1. Time
1.1.1.1. In the present
1.1.2. Example
1.1.2.1. I can run very fast
1.1.2.2. I can not very fast
1.1.2.3. Can you very fast?
2. Should
2.1. Obligation or recommendation
2.1.1. Example
2.1.1.1. You shouldn’t work so hard
2.1.1.2. You should study more if you want to win the exam
2.1.1.3. Should I eat meat or chicken
3. Ought to
3.1. Synonym of should
3.1.1. Example
3.1.1.1. He ought to quit smoking
3.1.1.2. I ought to leave now
3.1.2. Note
3.1.2.1. It is never used in interrogative phrases in American English
4. Would
4.1. Preference or polite question
4.1.1. Example
4.1.1.1. I would like to eat spaghetti
4.1.1.2. Would you help me please?
5. Might
5.1. Posibility
5.1.1. Time
5.1.1.1. In the present or future
5.1.2. Example
5.1.2.1. My mother might get angry
5.1.2.2. It might rain later
6. Could
6.1. Skill or possibility
6.1.1. Time
6.1.1.1. In the past or future
6.1.2. Example
6.1.2.1. I think it could rain later
6.1.2.2. I couldn’t sleep last night
6.1.2.3. You could write a book
7. Express
7.1. Posibility
7.2. Skill
7.3. Obligation
7.4. need
7.5. Permission
8. Particularity
8.1. Never change
8.1.1. Example
8.1.1.1. I can speak french
8.1.1.2. She can speak french
9. Must
9.1. Obligation or need
9.1.1. Example
9.1.1.1. we must go now
9.1.1.2. You must not drink
9.1.1.3. Must we get up early?
10. Shall
10.1. Suggestions
10.1.1. Time
10.1.1.1. In the future
10.1.2. Example
10.1.2.1. Shall we go to the movies or a museum?
11. May
11.1. Posibility
11.1.1. Time
11.1.1.1. In the future
11.1.2. Example
11.1.2.1. I may be late
11.1.2.2. May we come in?