Authentic Singer with Sphinx decals.
Now for the fakes.
You can see that the fake decals don't look "quite right" and have an odd reddish cast. The metal parts on the fake machines look too bright and shiny. If the machine is in a treadle base cabinet, the wood is actually cheap press board. If it is in a fake bentwood carry case that case will have silver colored metal latches. Those latches are not correct. Vintage cases locked with a key. But the dead giveaway is the logo badge located on the machine front, under the stitch length regulator. If it has a red "S" it's a fake. Vintage machines have a brass colored logo as seen on the authentic machine in the top picture. Here is a link to someone who is selling the reproduction machines. Take a good look at the photos so you know what to look for in a fake. Buyer beware.