Never heard the phrase “set tub.” I’ve lived in New York, Colorado, California, and Wyoming with parents from Pennsylvania and Ontario.
I’ve heard “laundry tub” and “laundry sink,” and I usually call it the “utility sink.”
I’ve lived in Virginia all my life, and have never heard of it. However I’ve only been in one house that had one that I can think of. I’d call it a “laundry room sink.”
My mother grew up in the hills of Virginia, and in our DC-area house, we called the basement laundry tub the set tub. I have long assumed it was an Appalachian phrase but I’ve never seen confirmation. I have also very rarely encountered anyone who recognized the phrase.
The term “set tub” is an Americanism, which first occurred in about 1890, corresponding with the emergence of indoor pumbing, reached the height of popularity in the 1940s, and has now fallen nearly out of use at all, with the introduction of automatic home laundry machines. There was one in my house in the 1940s, but I never heard the term, so I assume it was a regionalism, neither mid-western nor mid-southern.
From where you set your clothes to soak before wrenching them?
Never heard the phrase before.
Missouri reporting in…I have never heard it called anything but a slop sink.
Scary even in a zombie thread …
Cleveland with western PA roots, we call it either a “utility sink” or a “laundry sink”. If you swapped “sink” for “tub”, I’d probably know what you meant, though it’d sound a bit odd. A “set tub”, though, until I read this thread I would have assumed that it was some variant of the bathroom body-washing appliance (maybe something to do with whether it was built into the room or just sitting on the floor).
I’ve lived in NJ, Arkansas, Wisconsin, Florida, Illinois, never heard it before.
I think this is my first Zombie!! 
Definitely “tub” because these suckers are big. I’m talking at least a foot deep, maybe 18" and about 3 ft wide, usually divided into 2 compartments.
I’ve always heard it called a laundry sink. Typically very soiled clothes are soaked in it. Or cloth diapers are rinsed out before adding to the washer.
Concrete. Yes, I have some vague memory that it would have been made of concrete. And most of us have washers “outside the house”, right? 
It does sound kind of Appalachian to me, but I never lived in Appalachia nor do I have any relatives from there.
Never heard “set tub”. I grew up in NYS and probably would have said “utility sink” if I said anything at all, but now having moved here, it’s a “slop sink”.
OK, I’m going to have ask my siblings about this when we do our annual Christmas phone call.