On my Featherweights and older singers, the gears are exposed on the underside and they are very easy to grease. Clean the old off and a hair dryer will help by softening the old grease.
It is grease not oil and I would imagine that Singer or any other sewing machine brand would be acceptable. I would do some research and Google to see if you can't find more information. Maybe there is even a mainetnance manual for your machine.
I was very fortunate in being able to take a repair and maintenance class from an excellent man that had worked with manual sewing machines for many, many years. The class was inexpensive and one of the best investments I have ever made.