Seriously?

If my brakes fail, and the parking brake is the only thing that will keep the car from going over a bridge or into oncoming traffic, then you better believe I'm going to use it. I won't be thinking about speed limits. Yes, the car will skid if you apply it quickly.

If you don't know what the parking brake does, then I suggest you go out and try it.

It's called the "emergency" brake for a reason. If your brakes fail, it should be your go-to. 

Obviously, you want to apply it at lower speeds for better control. However, in an emergency, that may not be an option. If possible, downshift and slow (manual trans) or put it in neutral and hit the shoulder/grass to slow (auto trans). But definitely use it before things get out of control.

@mmj Had a good recommendation: if you don't know what an e-brake does, try it out in a parking lot around 15-20mph. It is literally there to save your life, so yes I would use it along with other defensive maneuvers.

Along with @mountainmanjoe I will also be gearing down hard to get some engine braking.

The term "emergency brake" is kind of a misnomer.  I would bet if you look in your owner's manual, they refer to it as a "parking brake", not an emergency brake.  Having said that, I agree with Mod Man and MMJoe that in an emergency situation, you should use whatever you've got to avoid an accident or injury.  It's just the responsible thing to do.  Besides, you can always rinse the cake out of your shorts later.

When I owned a car with a manual transmission, I used the hand brake to hold the car when I was stopped at an inclined interesection since I don't have 3 feet: 1 foot to hold the clutch down while stopped in gear, 1 foot on the brake to keep the car from rolling forward or backward, and 1 foot on the accelerator pedal to give it gas to lauch the car from a stop.

If you mean hand brake with lever and cable to rear drums or discs, that is far weaker than pressing brake pedal with your foot. This means stopping distance is longer.

It happened on 1999 Fiat Uno on its last inspection. The guy was checking brakes and pressed pedal harder and one of brake hoses exploded. When driving home I was driving at max speed 50 km/h and used side roads with less traffic. I was using brake lever so hard that I felt pains in my back a day later but it still did was not sufficient braking. Pressing pedal swiftly several times at once still made better job as long there was still some brake fluid in system.

For electronic brakes common sense tells me, computer wont't allow engage them if a car is not stopped.