Driving any automobile is nothing, if not an exercise in constant blind faith. Just the fact that any soccer mom in the world can take a 50mph curve without wondering what would happen if a tire blew, a ball joint broke or an axle-shaft snapped is testament to the pragmatic advantages of deliberate ignorance. Still, it never hurts to be prepared for any eventuality, especially if you find yourself in circumstances where breakage is likely.
Open Differential, Accelerating in a Straight Line
Most cars use "open" differentials, which route power to the wheel with the least traction. If you break an axle-shaft or constant velocity joint in a front-wheel drive car with an open differential while accelerating in a straight line, then power will simply flow into the broken axle and spin it. The engine will rev and the axle will spin, but the car won't move. The same goes for front axle-shafts in a 4WD truck, but if the axle tube itself breaks, then odds are that your truck will nose-dive into the road.
Locked Differential, Accelerating in a Straight Line
Many performance cars use locked, limited-slip or "torque biasing" differentials. Locked axles are just that; the left and right wheels always turn at the same speed. Limited-slip differentials can transfer some or all of the power to the wheel with the most traction -- the one spinning slower -- and torque-biasing differentials can transfer some, but not all of the power. With any of these differentials, the car will suddenly veer in the direction of the broken axle since the other wheel is the one doing all the pushing.
Open Differential, Accelerating Out of a Curve
If the axle-shaft is on the inside of a curve -- the right side for a right-hander, left for a left-hander -- while accelerating out of a curve, the vehicle will most likely dart inward toward the curve. Front-wheel drive cars and trucks in 4WD naturally exhibit a certain amount of understeer as the front tires struggle to both accelerate the car and keep it turning. Should the axle-shaft break on a front-wheel drive car, it'll quickly-turn into a no-wheel-drive and probably exhibit neutral handling. A 4WD truck or all-wheel drive car will suddenly turn into a rear-driver, which can result in snap oversteer and a probable spinout if the driver doesn't lift off the gas soon enough.
LS or TB Differential, Accelerating Out of a Curve
Front-wheel drives with a limited-slip or torque biasing differential will exhibit a somewhat more severe oversteer condition as power flows from the inside tire to the outside. A clutch-type limited slip may induce a severe oversteer condition and possible spinout, if it's powerful enough to transfer all of the engine's power to one wheel. The same goes for all-wheel cars and 4WD trucks with an LS or TB differential, but far more so.
Locked Differential, Accelerating Out of a Curve
Fully locked front differentials will act a bit differently than those that allow the wheels to spin at different speeds. This is especially true off-road, which is the only place you're likely to see anything with a locked front differential. Cars have differentials because the outside tire has to turn faster than the inside tire; if you lock the differential, the inside tire will always spin while turning and contribute little to no traction. Under these circumstances, the truck may break either the inside or outside axle, depending on the conditions.
If the inside axle breaks, all the power will go to the outside tire, inducing an understeer condition. If the outside axle breaks, that tire will gain traction while the inside tire continues to spin, inducing a neutral to understeer condition. One caveat though, a locked differential may do exactly the opposite, particularly if the truck lacks the power to spin the outside tire while turning, or spin the inside tire during a tight turn.