In the world of dirt tyres there are 2 keywords: Flotation and traction. While flotation means nothing to a road going vehicle it can be the key to success in many off road situations.
When you see an empty beer can swimming in a mud hole it's flotation that keeps it on the surface. It means exactly the same for tyres. A flotation tyre stays on top of the surface and does not sink in. If you sink in sand you will not get any traction at all. Flotation is achieved by keeping the pressure on the ground below the figures that terrain can support before it deforms. Some surfaces can go as low as 20 lbs per square inch or 1,3 kg/cm2. Of course these are very soft examples.
Those soft terrains can be sand, snow or even mud. The principle is the same: As long as you stay on top you can drive over it. A tyre burried up to the wheelwells will never be able to dig itself free. Some terrains will require a certain speed like vast sandy deserts. They are ok if you drive on them all day but once you stop the vehicle sinks right to the doors. So keep always a speed of 10-15 mph. Below: In mud driving competitions speed is everything. As soon as you give the mud enough time to slide out of your way you're trapped.
Flotation tyres are rare and most are not suited at all for tarmac. Examples are the Terra tyres from Goodyear, made for swamps and tundras. Semi-flotation tyres are a good general choice. By using a larger footprint you can decrease the weight per square inch. Running low tire pressures also widens the contact area. That's the reason why off roaders in Iceland use such large tyres - they keep them over the surface. An characteristic of flotation tyres is their sidewall construction. It bulges out to make the contact area even larger. Those become more vulnerable against rock damage but give much smoother movement through sand and snow.
Traction is the other side. You can have as much flotation as you want, it's of no use without a bit of traction. Traction is as important on snow as on sand or mud. Sadly different stuff needs different tyres- great. A mud tyre will dig you in sand as soon as you put the least trottle on. It's like gearing: If you want a gearing for highways it will not be very good off road. While you can change this easy by switching to low box you're stuck with your tyres. So they have to be a compromise.
Mud needs either a large agressive tyre or a skinny agressive tyre. Large tyres have more flotation so keep on the surface and provide adequate traction. Narrow tyres sink to the ground and grip it if the thread is agressive enough.
In general you can say the larger the negative profile (the lugs between the rubber profiles) is the better the traction in mud. The best all-out mud tyres are the Super Swampers or the 'Nobs', a profile like an hedgehog on steroids. Both are of limited durability on roads and problematic on wet and even dry tarmac. With a bit of less biassing towards bottomless mud and still good on tarmac are the BFG Mud-Terrains above.
MTs are excellent on trails and don't tear them up as tractor-type tyres. In fact it was easy for them to clear this obstacle even with the muddy rut. Not as noisy and slippery as the Super Swampers they have the drawback of less traction in the extreme stuff.
This otherwise standard Turbo Diesel 90 equipped with Super Swampers made it up this hill while I had to use the winch on half way up. This is however an extreme situation. The ground was so slippery you could hardly walk up.
Snow needs a different approach. Grip on snow is provided by many small edges biting - at least until the snow reaches the underside of the doors. I made very good experiences with the BFGoodrich All-Terrains in 265/70x16, the same as found as standard equipment on NAS Land Rovers (below).
They seem to plow through any snow trown at them as long as you have enough power to keep them turning. Luckily with the Vitesse engine this was not an issue. And yes, I made it through this deep snow on the second attempt.
In snow, you use snow chains. Full stop.
Sand is again an completely different stuff. In Sand you want to keep on top as much as possible as hard ground may be several hundred yards deeper. For normal driving you need a tyre with as few transversal and negative profile ribs as possible. Good sand tyres are wide to reduce the load per square inch. Paddle tyres are only designed for racing applications and should be avoided by standard 4x4's.
The above tyre is well suited for sandy grounds and is unlikely to dig in. Very popular sand tyres were also made by Michelin in their X-series. They are great on sand but wear like hell on tarmac and don't like low temperatures at all.