Most of the SouthBends, Logans and Sheldons you'll run into are going to be 9" to 11". To get above that in those brands, you'll need to find a later/newer model, which will be harder to find and will cost more if you can find one.
Overall, there are hundreds of brands of lathes out there, don't be afraid to buy one just because it's not one of the "popular" models. Yes, parts for some of these can be a problem, but the fact is, there's lathes made today that you can't easily buy parts for. Buy a good working machine, and chances are it won't need much more than bushings or simple repairs.
Several years ago, I picked up a Springfield, a well-regarded brand, that needed extensive work. I found out that parts for it basically don't exist. There aren't many in the first place, and I've never yet seen one parted out. I had to make a couple of my own gears, and modify a compound off another machine for it, among other things, but today it's a fine machine.
3 Phase is not the issue today that it once was. A VFD (variable frequency drive) is relatively inexpensive, runs a 3ph motor off single phase, and gives you fun stuff like infinitely variable speed, dynamic braking and so on. There's also rotary converters, or even static converters.
Honestly, even for a small home shop, a small 3-phase is practically
preferable these days.
ATW, Warner Swasey, Rivett, Monarch, LeBlond, Pacemaker and Rockford are all good names.
-American Tool Works, who also made Pacemakers, and LeBlond tend towards fairly large lathes. Great if you can find one (a 14" Pacemaker is basically the ultimate home shop lathe) but keep in mind they can be twelve feet long and weigh six thousand pounds.
Rivetts and the smaller Monarchs all tend to be on the expensive side- again, great if you can get one, but a good Monarch 10EE might start at $5K.
Warner & Swasey never made an engine lathe, they only made turret lathes. They're common and inexpensive, but poorly suited for engine lathe duties. They're great if you need to make sixty or a hundred identical parts, but awkward if you need to make a single part.
Threaded spindles aren't inherently bad, as long as they're a common size, like 1-1/2"-8 or 2-1/4"-8. The larger is preferable, both for strength and, as noted, the ability to take 5C collets. (Very handy- I'm personally a big fan of collets, and 5C is by far the most popular size for lathes.)
And finally, as Mush notes, condition is everything. A clean, low-miles and tight cheap Chinese import is better than an old American brand that's badly worn, damaged and rusty.
Ideally, you'd want a good, fairly common American brand, but don't be afraid of non-Asian imports like Voest, Boxford, Graziano, Emco or Schaublin.
I, personally, have an 11" Logan with a threaded spindle, and a 10" Sheldon with an L-00 spindle. The Sheldon is my main machine, and has done everything I ask of it.
Doc.