In honour of April Fool's Day, I offer my readers this semi-short summary of some interesting yet alternative ideas in physics that seem to attract a lot attention but very little serious research from academia (sometimes with good reason, sometimes not). In each case, they have been proven impossible or nearly impossible, and yet still generate mountains of literature.

Perpetual Motion: This idea goes back to the dawn of science, and is simply the idea of building a device which moves forever without adding any form of energy to it. Some are as simple as a marble that rolls along a track forever, some are more complicated machines. But as yet no man-made machine has actually achieved this goal.
The reason is that the laws of physics are quite strict about conserving energy. With very few exceptions (the big bang being the most obvious) the laws of physics dictate that you cannot get more energy out of a system then was originally put into it.
And that means that perpetual motion is the best you can hope for, but that any friction, air resistance, generated noises, imperfections in the build, loose parts, or even electric/magnetic/gravitational pull on other objects is going to slow down the machine. And once it can slow down once and lose some energy, it will do it repeatedly and will eventually stop. It may be very efficient and run for a long time, but to truly be perpetual motion it must run forever. 
It is not entirely impossible - in theory you could launch a probe into space and it would travel forever (although it would likely have to be launched well outside of any galaxies or other gravitational sources). So for all intents and purposes perpetual motion machines are forbidden by the laws of physics.

Free Energy: For people who don't want to accept that conservation of energy effectively bans perpetual motion machines, there is the slightly more radical idea of free energy. The idea of free energy machines is that you not only get perpetual motion, but that you can remove some of the energy for other uses. (It should also be clarified that in this case free energy really does mean energy is being produced. There are thousands of gadgets and plans on the internet that also use the term free energy to mean siphoning energy from other sources, such as solar power, wind power, or even draining the magnetic fields around high voltage electric lines! In this case, we mean producing energy without no external sources)
Unfortunately, as with perpetual motion machines this one cannot exist. The conservation of energy is not only well tested, but it has actually been proven to be true with very few exceptions (again, the big bang is about the only excluded case since the laws of physics may not have existed at the moment it happened). And so if you remove energy from the system, it will slow down and eventually stop producing energy. And so far every single free energy machine that has been tested has proven to generate less energy than was required to build it. They simply cannot exist.

Zero-point Energy: This theory/phenomena is far more interesting than the previous two, because it hasn't been proven impossible yet and does have some plausible connections with legitimate physics. There is an odd property of energy in quantum mechanics, in that when the energy of a collection of particles is calculated you find that the total energy depends on the number and type of particles, but it also contains a constant energy term that is present even in vacuum! Stranger still, each possible state of each particle contains this constant, so that the total energy of the vacuum becomes infinite.
This is quite an interesting result. Even when there is nothing present in a region of space, there will still be an infinite amount of energy. So then the question arises, can any of this energy be removed from the vacuum and made to do work? So far the answer is no. There have been experiments with the Casimir effect, in which two conducting plates held close together are pulled together by the pressure from the vacuum energy and which therefore means they gain energy - but the amount of energy required to create them and place them close together is far greater than any gains. Some people are also experimenting with Casimir wheels in which a wheel can be made to turn with the same forces, but as far as I know it still hasn't been successful (or even a solid prediction of a workable configuration).
So although zero point energy is tantalizing as a potential source of free energy pulled from the vacuum of space, as yet no one has even thought of a configuration that has stood up to review. 

Cold Fusion: Another sources of cheap energy, and one that has been a pariah of the academic world for over a decade. At very high energies and pressures (comparable to the inside of the Sun) small nuclei like hydrogen or lithium are collided with so much energy that they fuse into a slightly heavier element. But as they fuse, they release more energy into the system then they had originally - some of the internal energies of the nuclei are released. And thus at large experimental facilities on Earth, scientists have succeeded in generating fusion energy. Hydrogen is plentiful here, so such a reactor would be a very efficient and clean source of energy for the world - unfortunately the temperatures and pressures needed make fusion reactors too complicated and expensive for practical use.
But there have long been theories and speculation that some fusion reactions could occur at much lower temperatures and pressures. In a cold fusion reaction, nuclei would fuse together at more reasonable energy levels but would still generate excess energy. At a few points in history, professional and semi-professional scientists have claimed to have found such reactions, but as yet none of the reactions have been reproducible by other groups. Adding to the problem is the fact that if cold fusion reactions exist, the amount of energy generated in these experiments is tiny - so small in fact that most positive claims have later been found to be the result of other energy sources and not really fusion. Unfortunately the amount of false claims have been so high that most of the academic world no longer considers cold fusion to be a possibility. So maybe there is a very rare reaction that qualifies as cold fusion, but it is very unlikely to be discovered without a large investment into equipment that can detect truly minuscule energy output.
Anti-gravity: Finally we had an entry that doesn't include generating energy! The idea is simple - since electric and magnetic forces can be used to either attract or repel, then there should be a way to repel gravity as well. In fact, there are simple devices that can completely cancel out an electric field in a region of space, so perhaps there is a similar device to exclude gravity. In either case, it is a simple idea but so far no workable design has been either built or even predicted to exist.

The problem seems to be that where electricity has two possible sources - positive or negative charge - and can be made attractive or repulsive by selecting different charges, gravity has only one source - energy/mass. To generate a true antigravity field would require generating negative energy fields, and so far we can't do that on any reasonable scale. (there are speculations about whether removing zero-point energy, such as in the Casimir effect, would generate negative energy fields, but these are still too small to measure). So although anti-gravity is not necessarily impossible in the distant future, for now it cannot be achieved due to the lack of negative energy fields - but who can tell what the future will bring?

Trisecting Angles:  In fairness this is not a physics problem, but a mathematical one. However I am including it as an example of a problem which has a concrete proof and yet still garners a lot of false results. 
In ancient Greece, geometers did not use measurements and algebra but rather did their proofs with unmarked straight edges (ie rulers) and compasses (pointy things for drawing circles, not devices for getting lost in the woods). It was easy to divide an angle into two equal parts using only these tools, but mathematicians wondered for centuries if an angle could be divided into three equal parts in this way. To this day, mathematicians and journal editors get letters from amateur mathematicians with claims that they have succeeded in trisecting the angle.
The problem is that it is not only impossible, but it has been well proven that it is impossible for almost two hundred years. The proof is too complicated to include here, but it is presented in any good textbook on analytic geometry. No matter how hard you try, the angle can only be cut into two, four, eight, and so on. But inevitably generations of amateur mathematicians will continue to try, and generations of journal editors will continue to send out form letter rejecting their 'proofs'.
So there it is - my top five list of alternative physics ideas (and one mathematics idea) that get a lot of attention and no real progress towards reality. Some are known to be impossible, some are just unknown but very difficult to prove or disprove. But for the forseeable future, they will continue to be the bane of academic journals throughout the world ;)
NB: This article was originally written and posted on April 1, 2012, but due to a computer error it was reposted on April 23, 2012.