Surfergirl,
Not sure about your Singer, but I have 5 (?) machines that use the long bobbins, and quite a few that use the round bobbins with the same type of winder. None of them have lock washers on the back. I think greywuuf is right that there is a shoulder on the axle screw.
None of mine have ever come loose and I've never needed to take any apart that far for cleaning, so I'm not sure. Since yours is loose, it wouldn't be that hard to take it apart for a deep cleaning and make sure.
Joe