Some savers could potentially be sitting on accounts paying them zero interest, figures released by the City regulator show.
The Financial Conduct Authority (FCA) released the data as part of a package of measures aimed at helping people with cash savings accounts to get a better deal.
New rules have also come into force, meaning providers will have to offer easy-to-understand key information in an up-front summary box to help consumers compare savings accounts.
Firms will also have to clearly remind consumers about changes in interest rates when an introductory bonus rate comes to an end.
The FCA's "sunlight remedy" data on its website shows the lowest interest rates available from 32 providers of cash savings accounts and easy access cash Isas.
The figures, which show savings rates as of October 1, do not represent what every customer is earning, because they deliberately focus on the worst possible rate that someone might potentially get. Some customers on particular deals may get higher rates, depending on their circumstances and how they use their account.
The FCA's figures show that HSBC and its sister bank First Direct have savings accounts, which are closed to new customers, where the interest rate could potentially be as low as 0%.
But both HSBC and First Direct said these accounts normally pay a rate of 0.25% - and the 0% rate would only apply during a particular month if the customer made a withdrawal during that month.
An HSBC spokesman said customers had a range of motivations when opening a savings account, with quality of service and convenience also potential factors.
He said: "Our Online Saver does pay interest of 0.25% with no interest being paid only in a month there is a withdrawal, and over 70% of customers receive interest each month.
"This account is no longer available and we have made existing customers aware of a similar account that they could switch to that also pays interest in a month where there is a withdrawal, and it is easy for them to switch to that account."
A spokesman for First Direct said its E-Savings account also paid 0.25%, with no interest being paid only in a month when a withdrawal was made.
He said: "The vast majority of customers receive interest each month."
First Direct also said that the account was no longer available and it had made existing customers aware of a similar account they could switch to which also paid interest in a month where there was a withdrawal.
The FCA's website also shows that Barclays, Bank of Scotland, Lloyds Bank, and Nationwide Building Society have cash savings accounts that are closed to new customers that could potentially pay as little as 0.1% interest.
The Co-op has an account that is closed to new customers that could pay as little as 0.06%, while Ulster Bank has a deal paying as low a rate as 0.01% potentially.
Looking at cash Isa rates, NatWest has a deal paying 0.25%, while TSB has an Isa with a rate potentially as low as 0.05%.
Previous research by the FCA into the cash savings sector found that for many consumers, competition in the sector worth over £700 billion was not working as effectively as it could.
Christopher Woolard, executive director of strategy and competition at the FCA, said: "The new rules coming into force today will help consumers get the facts they need to make an informed decision about what to do with their savings.
"In a well-functioning market, providers should be competing to offer the best possible deal to consumers.
"Our sunlight remedy data shows that some consumers could be better off by opening a different account.
"One of our regulatory priorities is the treatment of long-standing customers and we want to see all customers benefit from competition and innovation in financial markets."
The FCA is also working to speed up the length of time it takes to switch savings accounts.
In December 2015, the FCA committed to publish an industry target for seven day cash Isa switching. Bodies representing banks and building societies have agreed that from next year, a minimum of 80% of cash Isa transfers will be carried out within seven working days.