Here are some examples from daily life:
1. I should do my homework instead of watching TV. (I am watching TV and I know I am supposed to do my homework)
2. I shall pass this exam (this is the future and it sounds quite formal or "British"; I will pass this exam sounds much better).
3. I have to get a job this month versus I must get a job this month (have to shows external pressure while must shows internal pressure as well) - "must" sounds too strong for most day-to-day usage. If you are in a restaurant, you tell the waiter "I would like to eat this and that" instead of "I must eat this and that" (which sounds rude)