So I know that Electrum has a minimum transaction fee of 0,05 mBTC. MultiBitcoin has a big one if I remember, but what about the other ones ? When I want to send some little amount and that I do not care about the time that the transaction takes, I would like to put less than 0,05 mBTC.
I believe that you can send any value of transaction provided the fee is appropriate for the size of the TX. I assume Bitcoin core would allow you to do something like this. You can also generate TXs manually and broadcast it to blockchain.