First, from our experience, carts (any 2-wheel vehicle) are not as stable as a 4-wheeler. This is because the axle is your pivot point, horse in shafts is your third point on the ground. So any movement horse makes causes the front go up and down on the axle while he moves along. Getting in and out of a cart will pull on horse to a degree, whether using front entry or rear entry to reach the seating. Horse needs to stand very well, when used with a cart for your getting in and out.
4-wheels is the most stable, has 4 points on the ground all the time, no horse action affects the ride as it will in a cart. Shafts are hinged to the axle, so the action of horse is lost by the time it reaches the hinge point.
Carts with detachable shafts will be found in newer vehicles, modern construction, mostly metal over wooden carts. There are various brand names out in use, that are easily taken apart and reassembled. As the User, it will be up to you to keep an eye on the pieces used to hold things together, checking often for wear on bolts, nuts tightened enough, so things don’t “rattle apart” from not being put back together correctly or wearing out. You need to replace those nuts and bolts more often, they wear with all that taking things apart. Metal of cart, both shafts and receiver sockets, may also develop metal fatigue, just snap on you in time. Kind of like bending a paper clip until it breaks, which is metal fatigue in action.
Personally, I don’t like the ride in a cart, too much back and forth in the seat while horse is moving. This can be changed with different horses, the kind of action they have, but you ALWAYS have some movement riding in a cart from the horse.
Carts to turn much shorter, so that can be very handy Trail Driving, over a 4-wheeler if you meet a downed tree, mudhole from the rain yesterday, where trail is not passable. It is up to the Driver, how they drive on a trail, how tippy a cart can be when getting one wheel much higher than the other wheel. Those little inexpensive carts with bicycle type tires are VERY tippy on rough ground. You CANNOT prevent tipping once they start going up. Their wheels also will fold on turns, not meant to take sideways torque. Rubber tires can come off the rims with turns, while bumps can make the tires go flat. But they are cheap, widely available, so folks buy them anyway. Air filled tires always have a chance of going flat, though the newer vehicles have heavier motorcycle tires on metal spoked (not bicycle spokes) wheels, are a bit tougher in harder uses. Rubber on steel wheels or wood spoke wheels, with no air can’t go flat! Ha Ha
Not sure what kind of outfit you haul with, but winch to pull vehicle into truck bed or trailer is a HUGE help in getting loaded. You just push the button, winch does the work while you steer. A winch was one of my best-ever Mother’s Day gifts!! I think the air horn was my favorite gift though, got quite a bit of use from that as well!
Sorry, can’t think of any names besides the Sprint cart by Frey, as about the most modern vehicle that disassembles. Lots of good comments on that vehicle, but it always depends on what you expect of it, how well it works for YOU. Our friend loves his, but uses it for conditioning and CDE things, not Trail Driving. There are used ones out and about too. if you want to shop around.
http://www.colonialcarriage.com/category.cfm?id=202&title=Sprint%20Driving%20Carts