I can speak of this from the motorcycle roadracing world; we most certainly have to deal with this, too, at least in the fore/aft plane, if perhaps not in roll.
During acceleration, assuming the vehicle is rear drive as all serious roadracing motorcycles are, you are not getting anti-anything on the front wheel (the front suspension will try to unload no matter what you do) but you are certainly getting anti-squat on the rear wheel. During braking, substantially all of the braking force is coming from the front, and the rear suspension is going to unload no matter what else you do, because of the forward weight transfer. So you have to understand that there are limits to how much you can "anti" anything.
Let's talk about acceleration first. On a sport or roadracing bike, there is a down angle from the swingarm pivot to the rear axle, and the various chain-pull and tire contact patch reactions act to create anti-squat. With the bike stationary, if you block the front from moving (e.g. have it against a wall) and you apply drive force by partly releasing the clutch in gear, the rear will try to go up. This counters the weight transfer rearward in normal acceleration.
Anti-squat is what plants the rear tire against the ground during acceleration without waiting for the forces to equalize through the suspension. This means that when the rider starts accelerating out of a corner, the rearward weight transfer gives extra grip right away, without having to wait for the suspension to compress. "Does it equalize over time" yes, but in that time, if you don't have that extra grip, the rear wheel has already started sliding. If you have insufficient geometric anti-squat (swingarm pivot too low), the rear of the bike will want to slide out during acceleration out of a corner, and the rider will complain of the rear sliding out (insufficient rear grip). If you have too much, it raises the rear of the bike too much which changes the steering geometry in a way that makes the front want to wash out (insufficient front grip). This WILL cost lap time if it is too far wrong, but it depends on the bike ... if the bike has 12 horsepower, perhaps it is not too important, but if the bike has 180 horsepower, you betcha it's important!
Anti-squat is also what keeps the ride height of the rear suspension in a consistent range with less dependence on whether the rider has the throttle open or shut. If the anti-squat is right, the rear suspension is also less likely to bottom-out during acceleration. The front is going to top-out no matter what you try, though.
Drag racing bikes frequently are lowered too much for this anti-squat to work ... and they often use solid rear suspension (no springs) ... as has been stated many times in the past, you can make any bad suspension work, if you don't let it. They don't have to deal with simultaneous cornering and acceleration, though.
Up front ... Different story. You cannot make a traditional sliding-fork suspension have anti-dive. There have been a number of alternative front suspension designs over the years; BMW has their "Paralever" system in production but evidently they have found that it was best to only have a low-ish percentage of anti-dive. I know that in roadracing, riders intentionally "trail-brake" into corners, shifting weight forward which compresses the springs under braking but this also steepens the steering head angle and shortens the trail, which makes the bike turn in easier. If the rider gets it right, the braking pre-compresses the suspension and it stays right near that same amount of fork compression as the rider turns the bike in (leaning the bike also increases the suspension loading) and simultaneously eases off the front brake. It's worth noting that BMW's most serious sport bike, the S1000R, uses conventional forks without Paralever - no antidive.
I suspect that the rear anti-squat situation will transfer over to car chassis as well since the same traction concepts apply, but obviously the steering situation is different. Still, large amounts of anti-dive up front pretty much imply that the steering axis (caster, trail) is going to be changing with suspension travel, and my suspicion would be that this is A Bad Thing.
"Anti-roll" in car chassis pretty much means having a high roll center. With independent suspensions, that also implies having a "jacking" effect - if you go too far with this, see "Unsafe at Any Speed", by Ralph Nader. (The first-generation Corvair was far from being the only swing-axle-rear-suspension car afflicted by this.) Beam-axle suspensions can tolerate high roll center without creating jacking if the geometry is right, but that doesn't mean they are free of bad side effects; notably, going over a one-wheel bump will cause a "side-kicking" motion. American cars in the 1960s and 1970s dealt with this (rear suspension only) by using really soft suspension bushings and tire sidewalls ... responsive handling was the furthest thing possible from their minds. Try that with modern low profile tires and stiff bushings - try rounding a corner on a washboard bumpy surface ... not good.
