Singer's 1st machine was the model 1, made for comercial sewing, followed by the #2 and #3
Singers 1st domestic machine was the "family" (also known as the turtleback) It was SO BAD that Singer offered 100% trade-in value and melted down the ones he got back. (REALLY HARD to find these days)
The Letter A or Model A was his first successful machine.
The horizontal bar in the last picture with the hole in the left end is the take-up. the big spring in the back controls how hard it lifts.