You should never have to pay any fees for keeping a savings account. It should the bank that pays you interest for keeping your money in their institution loan term. Whether you simply want to create an emergency fund or are interested in certificates of deposit, you should look specifically for a no fee savings account.
It's always a good idea to go with an online bank, as they tend to higher far higher rates than traditional brick-and-mortar banks. If they don't have branches to maintain all around the country, then they have more money to operate, which they are able to pass on to their customers.
Obviously, you'll want to check terms and conditions, as boring as they can be to read. A particular bank might promise "NO FEES!" in big letters, but if you look a bit closer, you might find that there are actually some fees that you might be charged larger on, or some sort of limitations on what you can do with your money or the number of transactions you'll be allowed to make.
Avoid any bank that actually tries to charge you just to open an account. While there are usually minimum deposit requirements to create an account, and maybe requirements for putting a certain amount in each month in order to maintain a high APY.
Signing Up for a No Fee Savings Account
The process of signing up for a no fee savings account should be fairly simple, with the option of funding your account via electronic transfer, wire, or mail in check. There might be different interest rate tiers, and the tier you fall into will depend on the minimum amount you are able to put into the account.
You'll definitely want to ensure that any bank you are interested in creating a savings account with is FDIC-insured. Your deposit accounts MUST be insured with the Federal Deposit Insurance Corporation so that you can have a peace of mind that the bank is trustworthy and that your money is going to a safe place.
When comparing rates, think about how "hands on" you are likely to be with your savings. While you should try and keep as much money as possible in your account for as long as possible, you never know when an emergency situation might pop up. Don't tie all of your spare money up in a single account. You should be able to access your funds if you need them without being charged any fee(s).
By far, the best place to look online for a no fee savings account is with CIT Bank. Get a high-APY and enjoy daily compounding interest. The process of starting an account is fast and easy, and you'll want to look into the "Savings Builder" Service.