All are correct, and all are used. =)
1 and 3 mean it would be a bad choice not to do the action of the verb.
2 and 4 mean "I prefer..."
1. I had better listen to your advice.
2. I would rather listen to your advice.
3. I had better stay at home than go out.
4. I would rather stay at home than go out.
(Which ones are right? Can we use all the expressions?)
2 answers
1 and 3 are advisable to do.
2 and 4 are your preference.
2 and 4 are your preference.