In the UK, the Bank of England has been floating the idea of negative interest rates and asking banks what they think. My guess is that they will rightly say it is a very bad idea.
Negative interest rates turn the normal workings of banking on its head. Instead of receiving interest, depositors have to pay to keep their money in a bank. Borrowers will be paid to borrow instead of paying interest. If banks want to keep money on deposit at the Bank of England, they will have to pay to do so - instead of being paid.
Savers have no incentive to save with banks and will pull their money out, which may lead to a run on the banks. They may respond by saving more and depress economic activity in the process. Negative interest rates may even stoke asset price bubbles that have done more harm than good.