I have accessed several accounts online over the years, and the one I feel the most secure with is Barclays.
To access my Barclays account, I first have to enter my surname and a 10 digit membership number (provided by Barclays and does not include my date of birth or any other memorable numbers). On the second screen I have to enter the last four digits of my debit card number. I then have to insert my debit card in to a small card reading machine (looks a bit like a mini calculator) and enter my pin. The card reader then generates a random 8 digit number which I enter on screen to access my account. Basically, for somebody else to access this account they would not only need my login details, they would also need the card reading machine, my debit card and my pin number.
To access my Barclays account, I first have to enter my surname and a 10 digit membership number (provided by Barclays and does not include my date of birth or any other memorable numbers). On the second screen I have to enter the last four digits of my debit card number. I then have to insert my debit card in to a small card reading machine (looks a bit like a mini calculator) and enter my pin. The card reader then generates a random 8 digit number which I enter on screen to access my account. Basically, for somebody else to access this account they would not only need my login details, they would also need the card reading machine, my debit card and my pin number.