Do you want to pay low or no fees, while receiving a high interest rate? Online banking is for you. There are no bank branches to deal with - everything is online. There's sometimes even better security too.
United States
In the USA, the Bank of Internet offers a checking account with a savings account's interest. As of January 2008, the rate for a Freedom Checking account is 3.34%. Their Regular Savings account earns 1.98%. BofI reimburses you for ATM fees on request.
United Kingdom
Smile offers a 4.75% interest rate (as of December 2007) on a savings account.
Ireland, Australia and New Zealand
RaboPlus New Zealand (by RaboBank) or RaboDirect in Ireland offers 4.3% on savings accounts. Australia's RaboPlus offers 6.9% and NZ offers even higher. Their Digipass is great security for your accounts.