Google "singer treadle sewing machine", and then click on "images". When you find a photo that seems to match your treadle, click on it. This is one way you can figure out the model number.
If you could post a photo here, someone here is likely to be able to give you a model number. There may even be a manual for it free online, but you need to know the model # first. I would not rely on the Singer serial #, as there are probably some mistakes in their database.