This table from USA Today
lists the monthly fees from the lowest-cost checking accounts at the 10 largest banks and 10 largest credit unions.
Out of the 10 largest banks, only PNC has a free checking account. Out of the 10 largest credit unions, 7 have free checking accounts. One of the 3 with monthly fees is PenFed. Its checking account requires a $500 minimum balance or direct deposit to avoid a $10 monthly fee.