If you want to try free motion quilting on a vintage machine you need a Singer 15. There were many minor variations over the years, for example, different stitch length levers, and on some the feed dogs drop. My treadle is a basic Singer 15K (K just means it was manufactured in Scotland, in the factory at Kilbowie). Singer 15s are easily recognisable by the tension discs, which are on the faceplate at the back.
The bobbin is held in the bobbin case directly below the needleplate. This is the central bobbin mechanism, and it is this method of stitch formation which allows the movement of the work necessary for free motion quilting.
Many other manufacturers made machines based on Singer 15s, in particular in Japan after the Second World War. These machines are equally suitable for free motion work.
For a good look at this type of machine, you may want to watch the video with the threading instructions.
So if you have a treadle or electric Singer 15, or another manufacturer's copy, and want to have a go at free motion quilting, give it a try!