What experiments have been done that confirm $E=mc^2$? What experiments have been done that confirm $E=mc^2$?
Are there experimental results that contradict $E=mc^2$? 
Or are experimental results consistently showing this famous formula to be true?
 A: There is a huge mass of experimental evidence that confirms the mass-energy equivalence. The clearest example that it happens is, of course, nuclear power, both in its explosive and civilian forms. If you want a detailed breakdown of all the experiments that corroborate it, I would recommend the entire archive of Physical Review C or a similar nuclear-physics journal.
If you want a clear-cut example, I think the best way to see things is via something called the mass defect. In essence, you weigh all the different nuclei and you divide their mass by the number of nucleons they contain. Somewhat surprisingly, the resulting mass average is not constant: a helium atom is less massive (by about $7\:\mathrm{MeV}·c^2$) than two protons and two neutrons out by themselves, so we say that helium has a mass defect of $7\:\mathrm{MeV}·c^2$, or a binding energy of $7\:\mathrm{MeV}$.

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In a huge number of nuclear reaction experiments, we induce nuclear changes so that e.g. one cobalt nucleus turns into something else. We weigh all the ingredients and all the outputs, which are typically lighter than the ingredients, and we also measure the energy of all the outgoing particles. And, indeed, the extra energy in the output has always matched the mass difference between input and output - there are no observed deviations from this rule.
A: Emilio's answer was also the first that came to my mind, but I was not quick enough to post. However, even more precise experiments come from particle accelerators, and similar devices.
https://en.wikipedia.org/wiki/Tests_of_relativistic_energy_and_momentum
The power of the magnets in the LHC is determined by the relativistic mass of the particles going through it, which is over 7000 times the mass these particles have at rest. If $e=mc^2$ was wrong, and they underestimated the mass, the particles would hit the outer wall of the accelerator.
http://lhc-machine-outreach.web.cern.ch/lhc-machine-outreach/lhc-machine-outreach-faq.htm
To answer the second question:
I am not aware of any experiment refuting $e=mc^2$. There may be problems near or in a black hole, where relativity slams into quantum stuff.
https://en.wikipedia.org/wiki/Criticism_of_the_theory_of_relativity
A: Another set of experiments which support $E=mc^2$ are Compton scattering experiments. The mass-energy of the electron is an important quantity in analyzing these events, and the results are consistent across a wide range of energies for the primary photon and scattering angles.  The energy of the secondary photon is given by
$$E_{\gamma '}= \frac{E_{\gamma}}{1+\left(\frac{E_{\gamma}}{m_ec^2} \right)\left(1-\cos\theta\right)} $$
EDIT: Here's a link to the Wikipedia version of the derivation, although any modern physics or nuclear physics text will have it as well.
