I presume that stability advantage would disappear as soon as you added the seat if you actually sit on it.
Yes, but having a seat adds a different kind of stability.
I haven’t exactly put that much time into it, but to me, the seat didn’t suddenly open up any possibilities that a “regular” BC wheel doesn’t have. If you added a brake maybe, but when you go that far, you may as well build a freewheel.
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@waaalrus just uploaded some old peg unicycle stuff :
I have never tried an impossible wheel, but from my freewheeling experience, I can imagine that a unicycle with pegs lower than the axle would be more stable, it is probably the machine that would make it easier to do very long coasts (severals km?). When I practise freewheel, I can coast up to 350m, I don’t have a spot to do more (note that on a normal unicycle, I don’t do 10m…).
Anyway, on a freewheel unicycle, although it is easier to balance with your buttocks off the saddle, I think that putting a little weight on the saddle can be useful. On a flat road it is less tiring and with experience you can certainly use this support to better control your balance.
Well I say that, but I prefer pedal-operated machines, at least we can keep going when the road climbs.
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Surely that’s a limit imposed by where you’re riding rather than the distance itself?
Or do you find yourself tired to a point of not being able to continue to coast after around that distance?
The impossible wheel shown in waaalrus’s video would be extremely easy to convert over to foot platforms below the axle. What I would do personally is look for some scrap aluminum of sufficient strength that is already bent to 90 degrees, then simply cut two pieces of it that could bolt to the sides of the fork, and extend below the axle where the 90s would extend to the sides and serve as foot platforms. Aluminum would be my preferred choice of metals because its light and can be cut quickly with common power saws.
If I couldn’t find any aluminum that was already prebent, it wouldn’t be too much of a task to fabricate similar footrests out of 1/8" aluminum plate. There would probably need to be some L brackets to reinforce it, but still an easy job.
It is still in our loft, it was very cool and easy to use. I should bring it out to play some time
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I’d love to have a go on it when you do!
I put my BC plates on as a test when I set up my peg unicycle (based on a drift trike front end). I don’t remember exactly why I didn’t like them. I may have felt that they were too likely to cause (body or property) damage. The pegs make a stable enough base. I only regret not recording a video in a skate park. Photo from 2015-03-13:
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I think what would really be interesting would be an IW with the bc plates and a handlebar instead of a seat. I would think that would allow you to really aggressively change your center of mass relative to the wheel.
I’m thinking someone with a lot more athletic talent than I have could really tear up a skate park with that.
It’s not impossible. That was never a good name or definition for this type of uni. Better to pick a less-specific superlative, such as a variation of “ultimate” to go with the idea of the Ultimate Wheel. Like the Spectacular Wheel. Or the Massively Difficult Wheel. 
In the US we called them BC Wheels. This comes from the newspaper comic strip “B.C.” by Jonhnny Hart. A cave-person themed strip, it featured many moments in the life of Thor, the guy who (usually) rode his solid stone wheel with the wood axle. But that name is only meaningful to people familiar with the comic. Even though Johnny Hart gave the unicycling community a nod from time to time, such as giving permission (informally perhaps) for images of Thor to be used in promoting unicycle events. Also, unicyclist Chaz Marquette performed in one or more live-action TV commercials for Monroe Shock Absorbers (for cars), in which he rode a single, stone wheel. 
Also I would contend that a BC Wheel with a frame and seat would not be a BC Wheel; it would be a more “evolved” version of Thor’s invention. The advantage of having a seat on one of those is, primarily, so you can sit down. Who wants to stand all day?
And while I’m on that subject, this is the reason I have absolutely no interest in an electric unicycle that forces you to stand up the entire time you use it. The only advantage to a standing uni is to make the vehicle smaller, which is great for storage and bringing it on the train or bus. If I get one, it’s going to be one with a seat. I like this one I got to ride in Spain:
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For a decade I’ve had it in my head that it was from The Flintstones, but from a quick search it seems all of the unicycles in that had seats and pedals.
In the UK I think we also call them BC wheels (or at least I do 
).
I never saw much use of a BC Wheel/Impossible wheel. You would need hills for that. On flat ground you can only roll a few meters and then you have to run and hop on, which if it is one with a seat, is more difficult than if it was just a steer like the one that unicyclemos shows. At least with a freewheel uni you can stay seated, make speed by pedalling and then sit still to roll onward.
Last year I had bought this toy, which is a bit like a unicycle, though it has 2 wheels next to each other, it is still self balancing and you can sit on it. However I got it refunded when I found out we are not allowed to ride it on the public roads in NL. It is too much money when you’re only allowed to ride it inside the house 
With pegs this is called an unbike, a phenomenon of the BMX world. Knowing that term you can find lots of posts and videos on this site and elsewhere. I discussed it on my thread Adventures in Peg Unicycling. @LargeEddie was an aficionado and I believe the picture of the gold-rimmed custom unbike on this page is his. I made and briefly rode this one (using the same drift trike fork as base that I used for my regular peg unicycle):
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I can coast over this distance in a 350m dead end near my home which has a gradient of about 2%. I have also trained in my street (200m at 3%) but it is too steep, the few times I have been down there I reached 25-30 km/h down the street. And the times when I didn’t reach the bottom, I risked damaging the cars parked in the street.
At the beginning, I found it easier to do the 350 m with a good speed at the start (20-25 km/h), but the last times, I tried to do the 350 m as slow as possible (15km/h, the average speed depends a little on my speed at the start, but mostly on the wind direction (and maybe bearings)). It is less risky and I think it is a better training.
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How do you start rolling with it? First with one foot on to find your balance and then add the other foot? And is it easy enough to hop off when you lose your balance?