Now, the main version (well, the only version) I've done is one where tori steps behind uke and places his arm in front, and then sits down along with uke. It makes for a nice, easy way to learn the throw for sure (although it starts to resemble tani otoshi).
For reference, the sempai in this one does it the way we usually do it:
Still, I wanted to learn a little more about it, so I started to do a little research.
It turns out that there about as many ways to do sukui nage as there are ways to top a pizza. The first knot I had to untangle is that there is actually two different throws often passing by the name sukui nage. There are versions of it that are also referred to as te guruma ("hand wheel") which, to me, are really a subtly different idea that I would prefer to keep distinct. Additionally, I don't like how te guruma variations seem to involve a good bit of lifting, which is fine if your opponent weighs the same or less than you, and you're young and buff enough to do it.
Consequently, I've decided to isolate for now just the versions which deal with stepping behind uke and scooping him backwards.
The video below demonstrates a lot of the variations you might encounter as te guruma. The latter half, incidentally, contains a nice section on variations of morote gari (which turn into techniques involving one hand, even though "morote" means "double hand"). I kind of like many of these because they don't involve lifting the guy. I'd like to look at them in class, eventually, but another day.
Anyway, back to sukui nage. Perhaps one of the biggest reasons we haven't spent much time on this particular throw (the classic version anyway) is the fall. For one thing, the mat at our dojo, while it has a little bounce to it, it's not quite as soft as the typical, pukey green rectangle mats most dojos use. So, if you grab uke's knees and dump him straight back, he'll end up landing square on his back and at the very least it will knock the wind out of him; at most, his head snapping back can definitely ring his bell.
Then I noticed some ukes taking a slightly different fall where they rotate a little and land on their side, more like a standard flipping fall (tobi ukemi). That eases my concerns about uke getting seriously hurt somewhat, but many of the students I work with aren't ready to take that fall, either.
This version, which looks suspiciously like aikido's gedan ate (the arm higher up, etc.), appears to be a little easier for uke to take, as only one leg is really scooped.
This version is another sutemi variation (where tori falls with uke), and looks a little gentler, similar to the sit-down version we typically do, but with a turning action. Uke still drops over the knee, but it seems less drastic (of course, they do have the softer mats).
[On a total side note, while I like many of the videos from this group, I can't for the life of me figure out why the chose blood red mats with crayon yellow walls, trimmed with black and red stripes. It literally makes my eyes hurt looking at it. It would drive me insane working out in that space!]
Ultimately, I'd like to flesh out a version that's both "safe to practice" as well as "effective and likely to find".