In OALD, the 3rd definition of might is used to make a polite suggestion.
You might try calling the help desk.
I thought we might go to the zoo on Saturday.
While the 5th definition of could is used to suggest something.
We could write a letter to the director.
You could always try his home number.
I wonder if both usage is common in everyday speech. Which is preferred or more polite? Are they interchangeable in making suggestions?