I got a Singer 401a yesterday. Mine does not zig zag - the part that should zig zag is full of 3 in 1 oil.
http://www.quiltingboard.com/vintage...3a-t70011.html on the second page of this posting there are pictures - this will show you what to clean and oil for the zig zag. There is another place near the camstack. You can see it when you move the needle bar back and forth.