I use little ziploc bags. I have tiny ones that I've saved from buying crystals and stuff, then supplimented with little cheap ones that I bought on Amazon. The little ones just fit the bobbin inside so that the thread doesn't unwind. Then I use bigger bags that I've saved from everything I buy and put the spool and the bobbin in the bag so they stay together. For cones, I slip the bobbin (in the little ziploc) inside the cone before putting the cone in another bag. Then the bag goes in one of my thread drawers.
This works well for me, but I don't have fancy electronic bobbins. I don't know what the issues are there.