Some banks and building societies have edged rates up on easy-access accounts or launched competitive fixed rate deals, but they are still at historically low rates and likely to remain there for the foreseeable future. Experts predict there is very little chance of a rise in base rate this year.
But with Post Office Online Saver, make sure you move your money after the first 12 months. The 1.01% includes a 0.76% bonus which only lasts a year. After than the rate plummets to 0.25%.
With French-owned RCI, your money is covered by the French compensation scheme which gives €100,000 (around £85,000) if the bank runs into trouble rather than the UK scheme, which covers you up to £75,000.
Other top deals
Virgin Money pays a slightly lower 0.95% on its Defined Access Saver 9, available online, through its branches or by post. You are limited to three withdrawals a year - make more and the rate drops to 0.25%.
Leeds Building Society pays 1.3% fixed until 2 April next year and gives you access to half your cash in the bond during the term without paying a penalty.
On easy-access cash Isas the top rate comes from National Savings & Investments Direct Isa at 1%. You can't transfer your existing cash Isas into these accounts.
The best deal for transfers is Virgin Money Defined Access Isa, available online, through its branches or by post, at 0.95%.
But on this account you are limited to making three withdrawals a year. If you make more, then your rate drops to 0.5%.
Family Building Society Market Tracker Isa pays 0.9% with no withdrawal restrictions and accepts transfers.
This story was originally written for our sister magazine, Money Observer.