Do you believe "E=mc^2" is a correct and/or useful equation, and, whether yes or no, precisely what are your reasons for holding this belief (with such a degree of confidence)?

Consulting only my memory: I believe it's the special case of a larger equation describing an object's total mass-energy, something like E^2 = [something based on the rest mass of an object] + [something based on the object's momentum], and when the object's velocity is 0, it becomes something like E^2 = m^2 c^4 + 0, which simplifies to E = mc^2.

I believe it's correct—let's say at least 99.9% confidence that this is what special relativity says, and 99% confidence that special relativity is accurate in some broad range of scenarios (though there's probably a "more correct" theory that will replace relativistic quantum electrodynamics eventually).  I believe it's useful for physicists—"it" being the full E^2 = ... equation.  If you mean the E=mc^2 special case... I think it may still somewhat useful for nuclear reactor engineers, because a noticeable amount of mass evaporates when nuclear fule is spent.  Maybe some other specialized engineers.

I believe the equation itself because I've seen statements to that effect (some more explicit than others) in a few different physics books and Wikipedia, and have never encountered a reason to doubt them in this subject.

For an object at rest (and let's assume we don't have to worry too much about gravity), $E=m{c}^2$ is a correct equation, where $E$ is the overall energy of the object, and $m$ is its mass, and $c$ is the speed of light. For an object that's moving, it also has momentum ($\overrightarrow{p}$), and this momentum necessarily implies that the object will have some kinetic energy adding on to its total energy. Special relativity provides a more general version of this equation that is relevant for moving objects as well. Namely:

$$E^2-(\overrightarrow{p}c{)}^2=(m{c}^2{)}^2$$

This reduces to the original $E=m{c}^2$ version if the object is not moving ($\overrightarrow{p}=0$). Another interesting special case is when $m=0$. This is the case with photons, for example, which are massless. Then the above reduces to:

$$E=c|\overrightarrow{p}|$$

So photons have energy proportional to their momentum. (Which turns out to be equivalent to saying that their frequency is inversely proportional to their wavelength. Which has to be true, since they travel at the speed of light.)

Note that in special relativity, energy is frame-dependent, and if you want to deal with quantities that are the same in every frame, you'll want to use the "4-momentum". So one other requirement for using this equation is to fix a frame where we're talking about the energy in that frame.

Source: Physics undergrad degree, several courses covered various aspects of this material. Part of this was learning about the various experiments that were done to establish special relativity. Back in the day, the Michelson-Morley experiment was quite a big piece of evidence, as were the laws of electromagnetism themselves, which were already very-well pinned down by Einstein's time. Now we have much more evidence, what with being able to accelerate particles very close to the speed of light in the LHC and other accelerators.

(1) Physics generally seems like a trustworthy discipline - the level of rigor, replicability, lack of incentive for making false claims, etc. So base rate of trust is high in that domain.

(2) There doesn't seem to be anyone claiming otherwise or any major anomalies around it, with the possible exception of how microscopic/quantum levels of things interact/aggregate/whatever with larger scale things.

(3) It would seem to need to be at least correct-ish for a lot of modern systems, like power plants, to work correctly.

(4) I've seen wood burn, put fuel into a car and then seen the car operate, etc.

(5) On top of all of that, if the equation turned out to be slightly wrong, it's unlikely I'd do anything differently as a result so it's not consequential to look very deeply into it (beyond general curiosity, learning, whatever).

As a personal convention, I don't assign probability something is true above 99% for anything other than the very most trivial (2+2=4). So I'm at 99% E=mc2 is correct enough to treat it as true - though I'd look into it more closely if I was ever operating in an environment where it had meaningful practical implications.

I believe it because physicists-as-a-whole seem quite sure about it, and I've never seen (modern) physicists be wrong about something they're this sure about, and frankly I know fuck-all about physics beyond what I learned in high school. 98-ish % confidence that it's as correct as any model of the universe reasonably can be.

I've studied physics, so I've gone through the proof.

Well, I've used tools that only work if special relativity works. And people seem pretty sure of it. So let's say 15 9s that relativity hasn't yet been detctably (to humans) violated on Earth - 99.9999999999999%.

Unsure whether quantum gravity will require violating relativity. Wild guess of P=0.6 that it will.

I am a sheep, I follow the herd. High-status people in my bubble believe that E=mc^2, whatever that might mean.

Add me to those who have been through the physics demonstration. So I'll give it odds of, let's say, 99.9999%.

But I also don't like how most physicists think about this. In The Feynman Lectures on Physics, Richard Feynman taught it as energy and mass are the same thing, and c^2 is simply the conversion factor. But most physicists distinguish between rest mass and relativistic mass. And so think in terms of converting between mass and energy. And not simply between different forms of energy, one of which is recognized to be mass.

But let's take a hydrogen atom. A hydrogen atom is an electron and proton. But the mass of a hydrogen atom is less than the mass of an electron plus the mass of a proton. It is less by (to within measurement error) the mass of the energy required to split a hydrogen atom apart. I find this easier to think about within Feynman's formulation than what most physicists do.

It’s useful in that it is a model that describes certain phenomena. I believe it is correct given the caveat that all models are approximations.

I did a physics undergraduate degree a long time ago. I can’t remember specifically but I’m sure the equation was derived and experimental evidence was explained. I have strong faith that matter converts to energy because it explains radiation, fission reactors and atomic weapons. I’ve seen videos of atomic bombs going off. I’ve seen evidence of radioactivity with my own eyes in a lab. I know of many technologies that rely on radioactivity to work - smoke alarms, Geiger counters, carbon dating, etc.

I have faith in the scientific process that many people have verified the equation and phenomena. If the equation was not correct then proving or showing that would be a huge piece of work that would make the career of a scientist that did that. I’m sure many have tried.

Overall the equation is a part of a whole network of beliefs. If the equation was incorrect then that would mean that my word model was very wrong in many uncorrelated ways. I find that unlikely.

I have very strong confidence that it's a true claim, about 99% certainty, maybe 99.9% or another 0.09%, but I am sufficiently wary of unknown unknowns that I won't claim it's 100%, as that would make it a malign prior.

Why?

Well, I'm not a physicist, just a physician haha, but I am familiar with the implications of General Relativity, to the maximum extent possible for a layman. It seems like a very robust description of macroscopic/non-quantum phonomena.

That equation explains a great deal indeed, and I see obvious supporting evidence in my daily life, every time I send a patient over for nuclear imaging or radiotherapy in the Onco department.

I suppose most of the probability mass still comes from my (justified) confidence in physics and engineering, I can still easily imagine how it could be falsified (and hasn't), so it's not like I'm going off arguments from authority.

If it's wrong, I'd bet because it's incomplete, in the same sense that F=ma is an approximation that works very well outside relativistic regimes where you notice a measurable divergence between rest mass and total mass-energy.