This Bloomberg article
reviewed the difficult environment that exists for savers. Here's an example it gave of a saver with $10K in savings looking at Chase:
Had they gone with the Chase savings account, it would have paid about $1 in annual interest. Potential fees included $4 if the balance fell below $300, $2 for each non-Chase ATM withdrawal and $3 for each withdrawal after making more than four withdrawals in a monthly statement cycle.
In my opinion, if you want a local bank, you should choose a bank or credit union with a free checking account. For a decent savings account rate, open an online savings account at an online bank like Ally Bank, Sallie Mae Bank or SmartyPig. Connect that free checking with the internet savings account. If you're lucky, you'll have a local bank or credit union that offers a free high-yield reward checking account.