Mary,
There are several different attaching methods. The most common is the one piece that screws onto the shaft with a thumb screw. Then there is the snap on. The snap on uses a spring loaded adapter that is mounted on the shaft and you snap the foot on and off of that part. The problem I've seen with the early Singer type with the narrow attaching point is they wobble and move around too much. I don't know about the pfaff designs. The newer designs use a 1/4" attachment and is more solid. I have very little experience with that design.
If your not familiar with the old one piece type look at my thread:
http://www.quiltingboard.com/vintage...t-t186875.html
Look close at post #1 and #10. You can clearly see where the foot pivots. Instead of a spring loaded point, there is a solid ankle and foot with a pin holding them together.
Joe