I am with ICICI bank for last 10 years. I have accounts with both AXIS and HDFC as well. My personal experience is ICICI is the best among the lot.
HDFC and AXIS have very poor customer service, very slow to get work done even small things like changing email id can take days, also they are always deducting small amounts from account, with AXIS even messages on mobile is a paid service and using net banking is paid because they force you to buy a security pack for online transactions (strange they force each customer to buy security which should be the basic responsibility of bank to provide to customers). ICICI is much better overall, no hidden charges and even if they deduct some amount, send an email to them online and in 90% of the cases they will revert the charges.
HDFC is very poor and rigid in reverting charges they deduct. Note - One important thing I would like to mention is I have never taken any loan from ICICI and I do not hold CREDIT card of bank. I have instant credit card against my FD, which is working very well. My review above is about regular banking with the bank, especially on private banking saving account I hold with them for last 10 years. I do not know how this bank performs on Credit Card and Loan side.