I am a huge fan of cardboard storage bobbins. They are relatively inexpensive, hold a good amount of yarn and I can write on them to keep track of my project.
I use my Hanson miniSpinner with the quill attachment as an electric bobbin winder to fill my storage bobbins. I have always been irritated that they don't fit - the bobbin openings are too big for the quill and don't spin without reducing the size of the bobbin orifice. I used to stuff fiber inside the bobbin to reduce the size of the opening and make the bobbin fit tightly on the quill, but I found that fiddly. For awhile I used rubber bands that I wound onto the quill, also fiddly.
I wanted a solution that was less fiddly because I frequently will take a bobbin off and weigh it in the midst of winding, if I am trying to divide the yarn from a large bobbin evenly between several storage bobbins.
One day I got irritated enough to take a trip to my local hardware store, quill and bobbin in hand. I found nirvana in the plumbing aisle. Rubber grommets. I couldn't find a grommet that both fit snugly on the quill and was thick enough to hold the bobbin in place. So I bought two sizes, one that fits tightly on the quill and one a size that is bigger than the storage bobbin opening. I group these on the quill with small grommets on either side of a large grommet, holding it in place.
Now I use my little grommets group on the quill on each end of the bobbin and it holds the bobbin in place every time, no fiddling.
Do you have a different solution?