In my experience with my two three wheeled go-karts, having the single wheel at the back is much more stable. I’ve never worked out exactly why that’s the case though.
It's because almost all vehicles can decelerate much faster than they can accelerate. Add to that that they may be decelerating into a corner and you can see how having a lone wheel in front is going to cause a lot more tipping over than having it in back.
The old-style three-wheelers (that they haven't been allowed to sell for decades) didn't have a differential in the back, so you had to tip them to the outside in order to turn sharply. Yes this was dangerous.
You didn't have to tip them. You just needed to get them to break traction on the inside wheel. You can do that without leaving the ground though leaving the ground is a pretty guaranteed way to accomplish that goal.
I’m not a physicist but I’m assuming because the most stable part of the chassis is at the front of the vehicle inertia is less likely to make it unstable. I’m sure, if you took those go karts in reverse to the same speed as a go kart with a single front wheel moving forward, that turning would be equally unstable.
As far as straight line stability I’m sure it has to do with being pushed vs pulled. I wonder if a FWD single-front-wheeled go kart would be more stable than a RWD single-front-wheeled go kart
The force of a turn spreads over the wide, instead of the narrow end, when you're driving towards the two wheels. You have to be more careful with trying to brake into a turn tho.
Polaris makes a "Slingshot" vehicle in that configuration, it looks like fun. From just looking at and sitting in one someone was trying to sell, I was concerned with its balance at speed. It felt light and as if it would be easy to put into an unrecoverable spin. Never got to drive it, alas.