The man of steel is impossible. Let's not pretend otherwise. The guy has X-Ray vision. He has super strength. He has supercold breath. He goddamn flies. There is no way he could do all that. But if there were, here's how they would work — according to what we currently know about the laws of physics.
Let's start with the physics of x-rays. These rays pass through any atoms without enough mass to stop them. When atoms absorb a photon, one of their electrons has to jump up from a space near the atom to a space farther away from the atom using a precise the amount of energy. The problem is, electrons can't just jump into any space – they can only occupy certain energy levels. If the atom that a photon hits doesn't have the energy levels to absorb the photon, the photon passes right through it. Lightweight atoms just don't have the energy levels to absorb high energy x-rays. In the human body, only calcium manages to really absorb x-rays. Since there's no calcium in the photoreceptors in our eyes, we can't see x-rays. If Superman can see them, then his eyeballs, when exposed to enough yellow sunlight, must incorporate calcium into their photoreceptors. In fact, Superman is shown as picking up a lot of different frequencies of light, and so has to have a lot of materials in the photoreceptors of his eyes, each sensitive to different kinds of electromagnetic radiation.
A tougher question is how in the world he is seeing these wavelengths in the first place. X-Rays are hard to come by on earth. In medicine, they're created by aiming a high energy electron beam at a target made of tungsten, usually in a vacuum tube. Either the electrons in the beam knock an electron off of an atom of tungsten, causing a higher electron to drop and fill its place, or the nucleus of the tungsten makes the electron beam veer off-target. Either way, the change in energy of the electrons produces a high-energy x-ray photon.
Now, it's been shown that Superman can produce high-intensity lasers with his eyes. In everyday life, that usually involves manipulating photons within a mirrored tube, reflecting them back and forth until they shoot out in sync with each other. The lasers Superman shoots have to be very hot and high-energy. Perhaps he has the ability to produce a chemical reaction energetic enough to cause x-rays to shoot out of his eyes, but they still have to bounce off a surface in order to get back to the photo-receptors in back of his eyes so he can see them.
If the x-rays are energetic enough, they may keep going until they reflect off some surface – x-rays will reflect off mirrors, but only at a shallow angle, so they'd have to reflect many times off many convenient mirrored surfaces – and some come back to him. This would expose whoever he's looking at to not one but two doses of radiation. It's probably better for humanity if he just is extremely sensitive to the few x-rays, and other cosmic rays, that manage to straggle through earth's atmosphere and magnetic field.
Not many people know this, but there are some stories that say Superman's invulnerability is not from inherent toughness of flesh, but from a superthin aura that surrounds him at all times. This is why the cape may get tattered in many battles, but no matter what the suit never comes off to leave a naked Kryptonian having on-panel battles with laser-toting aliens.
As everyone who has seen the trailers for Superman Returns knows, metal bullets bend on this aura, even when they're over the iris of the eye, but they are not repelled, so it's not electro-magnetic. And while not many people want to touch Clark Kent, those that do don't notice anything strange, so it can't be heat-related or a super-rigid material.
Plenty of people shed skin cells, and sweat during the day. Since Superman is made up of different materials (or at least his eyeballs are), whatever he sheds has to be easy for a slow-moving object to move through, but turn impenetrable on impact. It has to, therefore, be a non-Newtonian fluid. Many non-Newtonian fluids can be manipulated quite easily if they are touched slowly and without much force. A person standing on one would sink into it. However, give the fluid quick push, and it become rigid, and resistant to force. A person running on such a non-Newtonian fluid can make it across without any trouble, since the impact of their feet makes the fluid stiffen and support their weight. Likewise, someone slowly and carefully touching Superman wouldn't notice anything amiss, while a sudden impact would make his non-Newtonian fluid aura turn impenetrable. Interestingly, custard is a non-Newtonian fluid. This theory could be tested if anyone were to lick Superman and see if he tastes sweet. Volunteers?
Superman doesn't actually have super strength. Take a look at the comics. He stops planes by holding on to the nose cone. This means the entire weight of a falling plane is balanced on a section of metal approximately the size of a human being's hands. That's like trying to stop a person toppling over by putting a knife, blade up, in front of their chest. It won't end well. It's not strength that's keeping that plane aloft.
It's negative mass. Negative mass is a so-far hypothetical state of matter in which all the properties of a particle are the same, except their reaction to certain forces. A push on negative mass would cause it to go flying in the opposite direction. Superman must have the ability to temporarily turn parts of his own body, and objects touching it, to negative mass. If a plane is crashing, he would take hold of the nose cone – and create a web of negative mass throughout the plane. The force of the plane would push on this web of negative mass, causing it to resist, and slow the descent of the plane, all without breaking the plane into pieces. Superman could create these negative mass nets within himself, too, to provide internal structure of the proper strength – so he doesn't fall to pieces either.
This also explains why Superman's strength seems to increase and decrease depending on the power level of the foe he is fighting. Negative mass can only resist force that's applied to it. Final evidence for Superman's powers of negative mass control is the fact that a clone made with half his DNA, Superboy, has telekinetic control over the movement of any object he touches. It's clear that Superboy was able to refine Superman's abilities.
Negative mass would also explain Superman's power of flight – though not in the obvious way. Negative mass and positive mass are supposed to fall towards each other, so he feels gravity the same way everyone else does. However, he also feels air molecules all over his body. If he turned them to negative mass, they would push his body upwards. By contrast, if he turned the outer skin of his body to negative mass (or his aura), the air pressing back against it would cause it to push him forward. So Superman uses air pressure and negative mass to fly around planets with atmosphere.
This shouldn't work in space, except for the fact that tiny particles are constantly popping in and out of existence throughout the vacuum. Superman could use these particles the same way he used air molecules. It's also possible that Superman exerts some kind of energy which encourages the production of these particles in space, helping him fly faster.
This is a superpower that isn't used often, but it's one of the more straightforward powers that Superman has. Clearly, he uses changes in pressure to cool substances. It's been established that he has the ability to simulate super strength. He should have this inside as well as out. This means that his windpipe and his lungs can exert huge amounts of pressure on whatever he sucks into them. When pressure is added to a substance, the heat of that substance increases. When pressure is taken away, and the substance is allowed to expand - and sometimes change from liquid to gas - the substance cools.
Refrigerators use ammonia, which boils at negative twenty-seven degrees, to cool food. The ammonia is compressed into a liquid and then released into a gas, cooling it and its surrounding area. Superman could keep pockets of ammonia, or nitrogen, or some other element, compressed and stored in a part of his lungs at high pressure. When he takes the pressure off as he blows it out his mouth, the substance expands and cools, freezing whatever he aims it at.