Why do people not used walking on flat ice mostly fall backward when trying to? It's my experience that most people (including myself) fall backward on their behind when trying to learn to walk forward on flat ice and fall forward when trying to run on it. When walking or running (in the now "normal" way: heel first, ball of the foot last, contrary to the aesthetic walking of a ballet dancer and the same we do when walking backward; in fact, the more Natural way to walk is like the ballet dancer does; see the picture below; only with the advent of shoes, especially the modern back shock-absorbing shoes; just try running while putting the front part of your feet first on the ground; feels pretty good!) on a flat non-slippery underground you mostly fall forward and mostly after tumbling (tripping sounds a bit inappropriate in this case). By the way, by walking and running I mean(in the now "normal" way: heel first, ball of the foot last, contrary to the aesthetic walking of a ballet dancer and the same we do when walking backward. In fact, the more Natural way to walk (which is now used to walk as silently as possible) is like the ballet dancer does (see the picture below). Only with the advent of shoes, especially the modern back shock-absorbing shoes we changed our walk. Just try running while putting the front part of your feet first on the ground. Feels pretty good! But let stick to the modern way. 

In can understand why, when walking on high friction underground you mostly fall forward after tumbling. For example, bumping with one foot against a curb you didn't see causes a torque (the force of which grips my foot in backward direction) making my center of mass rotating forward around my foot whereafter gravity makes me fall.
When walking on flat ice, only slipping is involved though, without obstacles your feet can bump against. Of course, there has to be some friction (and there is some friction, though little) because without friction walking or running would be impossible.
Now when you are an experienced ice walker, suppose your mass rests on your whole left foot, and the ice gives just enough friction not to slip (let's say in any fall of the normal walking on ice). Then you give your right foot forward motion. This forward motion is what gives you a forward motion in combination with the left foot that stays put but "rolls" over the ice and pushes you forward. (the movement of you right feet gives the left foot a backward force on the ice, which propels you forward. Meanwhile, the left foot changes from staying flat on the ice to standing on the front feet and when the heel of your right leg touches the ice it "rolls" along and your whole mass rests now on your right feet, after which the left leg starts moving and the cycle is complete. Nothing different from normal walking on a rough underground.
But why do people learning to walk on the ice fall so often backward, one of their legs moving upwards in a forward direction? Where can't the friction stop the slipping?
Of course, they also fall forwards but this is mostly the case when they try to run on the ice. 
 A:  
This is how our legs look. When we are standing our weight is balanced by our foot and heel, but we have an extended foot area which can produce torque if needed(in case we are about to fall forward) while no such support is present while we are falling backwards as the heel right on top of which our center of mass lie is the very end and any torque produced by gravity while tilting backwards can not be countered normally and we fall.
A: It seems to me:
When you are walking, and you suddenly feel a foot slipping, and you feel you are in danger of falling forward, you have a shot at catching yourself by kicking a leg forward. Automatically you kick out the leg that isn't bearing weight. Kicking out one leg is a reflex that can be executed very fast, I think. You may fall sideways instead, but I think you can almost always catch your self from falling forward.
Conversely, I think that at the instant you become aware you've started falling backwards it's already too late. Kicking out a leg backwards is much harder, and you can't reach as far. 
A: The first thing that should be asked is: "Why should I fall in the first place?" The answer being that there was (almost) no force (here static friction) to stop the relative motion between your feet and the ground. Now when you move your feet forward your upper body moves backward (so as to conserve the angular momentum) now since (almost) no external force acts on your body to generate the torque and hence give you some forward angular momentum. Therefore your body continues to move backward and hence falls. 
A: This is not theoretical question about physics as much as biomechanical question about technique of walking. 
When I run during winter, I always fell forward (assuming there is no slope). That is because I am putting my legs on the ground with backward motion to use the force to accelerate. So when they slip, they go backward, while my body center goes still forward.
A lot of people during walking are putting their legs on the ground with forward motion, so when they slip, the legs go forward faster than their body center and they fall on their backs. This is mostly because nowadays we learned to walk through our heels instead of toes as a consequence of wearing shoes. 
A: When we walk, rather start to walk we apply force on the ground in backward direction so the floor by Newton's 3rd law applies an equal force in the forward direction to keep one moving as one moves , to keep one from  falling the body applies a torque in the opposite direction to keep you from falling but as you enter a slippery area the torque by friction reduces but if the torque applied by the body before is not reduced, that leads one to fall in the direction in which the body applies the torque, the converse may also be true if the friction applies more torque one may fall in the opposite direction, Thus it depends based on which case you are suggesting.
A: Whilst there is plenty of sound physics in the other answers, I'd like to offer a linguistic alternative. We mostly fall backwards when we slip and we mostly fall forwards when we trip. These are convenient words that have evolved to describe different mechanisms of humans falling. Slipping is generally where your foot slides from under you, something which is more likely to happen with weight on the back of your foot as discussed in other answers and normally results in landing on your backside. 
So in summary you could argue that we slip backwards almost by definition.
