Brick-and-mortar banks such as Bank of America, Chase and Wells Fargo typically charge a monthly fee for a savings account. Monthly fees are usually waived if you maintain a set minimum balance. If you can't satisfy that requirement, consider a savings account without monthly fees such as Capital One's 360 Savings account.
Monthly fees can severely cut into your savings. For example, if you save $496 with Bank of America in your first year, you'll earn just three cents in interest by the end of the year, all while paying a total of $96 in monthly fees. Bank of America's monthly fee is higher than Chase and Wells Fargo, which costs $5 per month.