Banks should provide evidence of the quality of service - such as the frequency of security incidents - to allow customers to compare accounts.
The time taken to open an account or to replace a lost or stolen card should also be published, the Financial Conduct Authority (FCA) said.
Such service guides should make it easier to compare accounts, the FCA said.
The industry welcomed the proposals, which could take effect next year.
The proposals also cover business current accounts, and suggest quarterly publication of a string of service guides, such as how and when customers can make payments, cancel cheques and whether 24-hour help is available, or the amount of time before access to a current account is available through power of attorney.
The frequency of security incidents would be closely watched as, at the moment, there is little evidence available abouit the number of times bank services are unavailable due to security or operational incidents.
A spokeswoman for UK Finance, which represents the major banks, said: "Banks work hard to ensure the products and services they offer meet customers' needs - this initiative is a positive step that will make it easier for consumers and businesses to compare the quality of service offered by different current accounts.
"We look forward to working closely with the FCA to help ensure this information can be presented clearly, simply and consistently."
Gareth Shaw, money expert at consumer group Which?, said: "It is vital that consumers have access to useful information that helps them make informed choices about which bank to move to. However, many customers that would benefit the most from switching banks, such as those who regularly rely on overdrafts, don't switch.
"This is why we are calling on the FCA to use its review to end excessive changes for unarranged overdrafts."
Consultation ends on 25 September, with a proposal that figures should be collected from April and first published in mid-August 2018.