Can spinning arms really help you balance on the edge of a cliff? 
This is how many cartoons depict a character about to fall from the edge of a cliff. You can see movie characters do it. You can also see real people doing it. But does spinning arms like that actually help you to gain balance?
It would, ( through conservation of angular momentum) if your arms were like flywheels connected to a motor through an axle.

But they are not. (Try spinning your arms once without letting them flip in their own axis). As your hand completes a revolution about your shoulder, your arm completes a revolution about its own axis. It seems like things are not that trivial here.
Short, concise answers are most welcome, because this question has the potential to save many lives. And if you happen to be on the edge of a cliff while reading this, you are encouraged to perform confirmatory experiments.
 A: This is just expanding a bit on the answer of g s.
Anyone saying that arm spinning cannot change the (angular) momentum should first remember that it is possible to swing on a swingset.
Here is a simplified model that I think captures the main phenomenon (but I don't know for sure). The muscle group here is a stand-in for the psoas (hip flexor), though of course a lot of other muscles are involved. We consider the center of mass of the entire body excluding the arms, and argue that there is a net backward force on it, as well as other forces.

*

*The arm swings as the leg muscles tense. This produces a net backward force with no or small change in angular momentum.

*The arms reset, producing a small net forward force, as well as some forward angular momentum.

 
 
A: Mostly it's the first half swing which helps
There's a reaction mass thing happening with moving the mass of your arms compared to the mass of your body. The force required to move your arms forward applies an equal and opposite force backward on your torso. This will tend to pull your torso back over your feet.
Whilst your body is leaning forwards, your friction with the ground is reduced, and your natural leg movements will tend to push you forwards. If swinging your arms is enough to bring your torso back upright, then friction is improved and your natural leg movements will push you backwards.
Of course if it's not enough then over you go! But there is enough success rate for it to be a good strategy. Learning to use your arms as reaction mass for balance is one of the basic things we develop as babies when we first start walking, so it's a very well-developed reflex. It's not innate (as everyone who's seen a baby learning to walk can testify), it's just something we've used continuously from before we could talk, so the skill becomes reflexive.
Note that this relies on there being a solid surface underneath you, otherwise your equal and opposite reactions will equally well push your legs the wrong way. These reflexes are something you have to consciously unlearn when you start any sport which relies on you standing on a wobbly surface, such as windsurfing or paddle-boarding. For these, you need to develop a different balance reflex where you move your hips to move your centre-mass, pivoting around your ankles. This is actually a better balance technique for when you're stood still, because you're directly moving your body mass, but it doesn't translate well to walking motions.
After that, there will be a measurable effect, but not much
The OP claims that because the arm rotates about its axis as well as about the shoulder joint, it cannot produce a net forward force. This is clearly incorrect though. The net force for continuous arm spinning is of course produced by displacing air, so all we need is more air to be displaced on the back-to-forward part of the spin.
If you try the arm movements for swimming backstroke, you will clearly see that your arm and hand rotate to present your palm, giving maximum cross-sectional area through the air on the half-stroke from behind and passing your waist. As your arm comes back on the other side, you twist your wrist quickly so that your little finger is upwards. Your hand's cross-sectional area through the air is now reduced to only the blade of the hand, and it remains so for the rest of the rotation.
In a more dense medium, this would clearly have significant effects. You absolutely can do backstroke underwater to demonstrate the principle (although of course there are easier ways to swim underwater). In air though, you would need to be spinning those arms really fast, because you need to displace enough air to hold yourself up.
How much? Let's do the sums...
The worst-case of course is countering your entire body mass, in which case you're basically a human helicopter, and then you don't need to worry about falling because you can fly. Let's assume the effect isn't quite that awesome, so you can only support a quarter of your body mass with whirling arms. For an average 80kg adult, that means we need about a 200N force.
Looking at my hand, I can probably cup a half-litre of air, which (at an average air density of 1.2kg/m3) weighs 0.6g. Say 0.5g in round numbers, and for simplicity let's say (because you have two arms) that you're moving 0.5g of air continuously forwards. Let's also simplify the movement and say that you're accelerating this air linearly over a 1m linear distance, and that your arm movement has a 4m circumference (which gives you the speed of your arm, and hence the air). Then for revs/s R and F=ma,
200 = (0.0005 * R) * (((4*R)^2)/2)
8R^2 + 0.0005R - 200 = 0
Solving (and ignoring the negative solution),
R = 80 revs/s
So for your cartoon character capable of whirling their arms like helicopter blades, this could genuinely be a practical strategy. For humans, not so much.
A: The force of gravity pulling you down acts at your center of mass.  The sum of the external torques equals the change in angular momentum.  If you have no initial angular momentum gravity pulls you over (like a stationary bike turning over).  If you have some initial angular momentum gravity tends to turn you (like a bike rolling forward).  But I doubt you can create enough initial angular momentum in the correct direction to make much of difference (like a slow moving bike).
An analogy is jumping upwards from an elevator in free fall that has a large velocity downwards.  The decrease in your net velocity downwards is too small to save you.
Moving your center of mass (e.g. leaning away from the edge) will help, but this is differnt from swinging.
A: I haven't yet seen an answer that clearly states the conservation law in play, or clearly describes the forces of interaction with the environment.
Assuming a simple point contact with the edge, you cannot produce a torque at that point. Thus, the forces you can create do not affect your angular momentum around that point. Only gravity, which acts on your center of mass, does.
Thus, your angular momentum around the edge will increase unless you can get your center of mass behind the edge. To move your center of mass, you need reaction forces with the environment. What forces are at your disposal?
The contact force can be broken into two parts — a normal force and a tangential force from friction.
Ignoring the exact direction of the normal force (up and maybe a little forward), we do know it has to be positive (you can't pull on the edge.) The tangential force is perpendicular to the normal force, and has to be within the limits of friction, or you will slip. Intuitively, it seems like you might be able to create some horizontal force to move your center of mass backwards.
How can arm rotation connect to center of mass motion? Revisiting our "conversation law", angular momentum around the contact point is the sum of angular momentum around the center of mass, and the angular momentum of the CoM around the contact point.
That means that if there's more angular momentum around the center of mass, there's less motion of the center of mass around the contact point. Enough "internal" angular momentum, and the center of mass will move the other way.
That change in center of mass velocity has to be created by the contact forces. In other words, attempting to accelerate your arms will create a reaction force with the ground. That's what actually accelerates your center of mass.
But you have to keep moving them: only so long as you have some angular momentum around your center of mass can you keep your CoM moving back towards safety.
A: The human body is a complex multi-part machine. In this case, windmilling your arms does work (as part of a more complex multi-stage full body motion), and more than the initial forward swing of the arms is useful. It works sort of like a ratchet.
The body can change shape. Static friction between the feet and the ground can be used to produce a variable net rearward force on the body. The magnitude of the rearward force varies based on the body's shape, and applying the rearward force requires the body to change shape (specifically, straightening flexed legs). If the person is able to have their butt behind their toes, they can probably recover even if their center of mass is forward of their toes.
The order of events is like sequential backwards hops, except without leaving the ground. (If you try doing sequential backwards hops without leaving the ground, you will fall over backwards, which is the opposite torque from falling over forwards, which is the point.)
The person flexes the legs (negligible net horizontal force excluding gravity).
The person thrusts the arms forward, then immediately extends the legs (net rearward force excluding gravity).
The person flexes the legs while thrusting the arms back (negligible net horizontal force excluding gravity, because the legs and hips are moving forward while the arms are moving back - all forces are internal).
Here is a crude diagram of this 'ratchet' that I drew in MSPaint.


Edit for clarity: the net torque comes from the legs pushing on the ground. The arms just help position the body such that the legs can push in the desired direction (or, as SomeGuy points out in the comments, creates an internal torque that the legs and other muscles in the body can turn into a push in the desired direction by holding rigid during part of the motion). If it weren't for pushing with the legs, the person would (absent gravity) just wobble backwards and forwards as their arms went forward and back. Thrusting the arms forwards allows the hips to move momentarily backwards, during which time the legs can apply a greater horizontal force.
A: The cartoon version does exaggerate, of course.
Here is what I think: you only have time for one swing of your arms. If that first swing does not save you then by the time you go around a second time it's already too late.
Your sense of balance knows those things. Let's say you are on an edge, and it is half a meter down. You try one swing, and when your sense of balance feels that was not enough then your sense of balance will decide you must jump of the edge. By jumping your motion is controlled and you will land on your feet. If your sense of balance would not make you jump then you would fall over, and likely you will be injured.

I think there is a way to test this. You will need someone to assist you.
You need an edge with a very small height difference, for instance the height difference between road and sidewalk.
Stand with your feet on the edge of the sidewalk, with your heels unsupported (Comparison: the way competitive divers position their forefeet on the edge of the diving board before the jump.)
At that point the person assisting you should push you just a little. In order to be unprepared for the push I expect it will be necessary for you to close your eyes. (If your eyes are open your sense of balance can see your assistent make a move, and your sense of balance will commence countermeasures in advance.)
Under those circumstances: I expect that if it is possible at all to remain on the sidewalk a single swing will do it. If that single swing is not enough your sense of balance will sense that, and you will step down onto the road.
I expect that the help of an assistent is absolutely necessary; your sense of balance will simply disallow you from putting yourself out of balance.

As to the physics:
The physics is the same as the physics of a device that maintains balance by way of rotation of reaction wheels. Youtube video: Self balancing stick
Your feet are your point of contact, and that point of contact does not slide, so it acts as a fixed pivot point. So when the reaction wheel is spun up the stick is reoriented with respect to that pivot point.
A: Assuming you are in an unstable stance where your centre of mass is slightly past the brink - if you do nothing you will topple forwards and fall over the brink.
Throwing your arms forwards does not help at all!
Your centre of mass stays in the same place (conservation of momentum), if it was past the brink it is still past the brink you have gained nothing.
When you start spinning your arms that puts rotational momentum into them this puts a brief torque (or an anglar impulse) on your body via your shoulders which angular impulse may be enough to tilt you back onto the safe side of the brink.
When you stop spinning there will be an opposite impulse so be sure to full regain steady balance first.
Your arms don't twist when you spin them but that's not actually a problem,  rotational inertia is mass × radius squared, and you still have that.
You can experiment with this at a swimming pool or over a soft mattress. or hay pile etc.
