Are vacuum fluctuations really happening all the time? In popular physics articles and even some physics classes I've been to, the vacuum of space is described as being constantly full of quantum fluctuations.  Supposedly, all sorts of particle-antiparticle pairs at all scales are constantly appearing and disappearing.  We end up with a mental image of the vacuum as a roiling, choppy sea with all sorts of things going on, rather than a calm, placid background.
However, the vacuum, being the lowest-energy state of a theory, should be an energy eigenstate—which means it is time-invariant (except for a physically-irrelevant phase factor).  So it seems the vacuum really should not be seen as a dynamic entity with all kinds of stuff happening in it, as we're led to believe.
Jess Riedel wrote in a blog post that

A “vacuum fluctuation” is when the ground state of a system is measured in a basis that does not include the ground state; it’s merely a special case of a quantum fluctuation.

So it sounds as if the existence of vacuum fluctuations is contingent on measuring the vacuum—in particular, measuring something that doesn't commute with energy (such as, I guess, the value of a field at a point).
How much truth is there to the idea that vacuum fluctuations are constantly happening everywhere, all the time?  Is that really a useful way to think about it, or just a myth that has been propagated by popularizations of physics?
 A: It's true that the vacuum ought to be an eigenstate of the full interacting Hamiltonian. But as seen from the perspective of the Hamiltonian of the free theory (all interactions being treated as perturbations around this free theory) the actual ground state is "dressed" by many vacuum fluctuations on top of the free ground state.
A: Particles do not constantly appear out of nothing and disappear shortly after that. This is simply a picture that emerged from taking Feynman diagrams literally. Calculating the energy of the ground state of the field, i.e. the vacuum, involves calculating its so-called vacuum expectation value. In perturbation theory, you achieve this by adding up Feynman diagrams. The Feynman diagrams involved in this process contain internal lines, which are often referred to as "virtual particles". This however does not mean that one should view this as an actual picture of reality. See my answer to this question for a discussion of the nature of virtual particles in general. 
A: I think it’s possible to give a beginners guide to what is meant by vacuum fluctuations, but it necessarily involves taking a few liberties so bear that in mind in what follows.
Before we start let’s remind ourselves of the following key point about superpositions. Suppose we have an operator $\hat{n}$ with eigenfunctions $\psi_i$ and we place it in a superposition:
$$ \Psi = a_0\psi_0 + a_1\psi_1 + \, … $$
Then when we do a measurement of the system using our operator $\hat{n}$ the suprposition will collapse and we will find it on one of the eigenstates $\psi_i$. The probability of finding it in that state is $a_i^2$.
Now suppose we do a measurement, then put the system back into the same superposition and do a second measurement, and keep repeating this. Our measurements will return different results depending on which of the eigenstates the superposition collapses into, so it looks as if our system is fluctuating i.e. changing with time. But of course it isn’t - this is just how quantum measurement works, and we’ll see that something similar to this is responsible for the apparent vacuum fluctuations.
Now let’s turn to quantum field theory, and as usual we’ll start with a non-interacting scalar field as that’s the simplest case. When we quantise the field we find it has an infinite number of states. These states are called Fock states and these Fock states are vectors in a Fock space, just as the states for regular QM are vectors in a Hilbert space. Each Fock state has a well defined number of particles, and there is a number operator $\hat{n}$ that returns the number of particles for a state. There is a vacuum state $\vert 0 \rangle$ that has no particles i.e. $\hat{n}\vert 0\rangle = 0$.
Suppose we consider a state of the scalar field that is a superposition of Fock states with different numbers of particles:
$$ \vert X\rangle = a_0\vert 0\rangle + a_1\vert 1\rangle +\, … $$
If we apply the number operator it will randomly collapse the superposition to one of the Fock states and return the number of particles in that state. But because this is a random process, if we repeat the experiment we will get a different number of particles each time and it looks as if the number of particles in the state is fluctuating. But there is nothing fluctuating about our state $\vert X\rangle$ and the apparent fluctuations are just a consequence of the random collapse of a superposition.
And by now you’ve probably guessed where I’m going with this, though we need to be clear about a few points. The free field is a convenient mathematical object that doesn’t exist in reality - all real fields are interacting. The states of interacting fields are not Fock states and don’t live in a Fock space. In fact we know very little about these states. However we can attempt to represent the vacuum of an interacting field $\vert \Omega\rangle$ as a sum of free field Fock states, and if we do this then applying the number operator to $\vert \Omega\rangle$ will return an effectively random value, just as it would do for a superposition of free field states.
And this is what we mean by vacuum fluctuations for an interacting field. There is nothing fluctuating about the vacuum state, however measurements we make of it will return random values giving the appearance of a time dependent fluctuation. It is the measurement that is fluctuating not the state.
I’ve used the example of the number operator here, but it’s hard to see how the number operator corresponds to any physical measurement so take this just as a conceptual example. However the process I’ve described affects real physical measurements and happens whenever the vacuum is not an eigenstate of the observable measured. For an example of this have a look at Observation of Zero-Point Fluctuations in a Resistively Shunted Josephson Tunnel Junction, Roger H. Koch, D. J. Van Harlingen, and John Clarke, Phys. Rev. Lett. 47, 1216 available as a PDF here.
A: Vacuum fluctuations do exist, but they are not a statement about the dynamics (the time evolution) of the system. This is true for quantum fluctuations in general. The state of the system may very well be stationary, still quantum fluctuations will be present. A more correct statement is: quantum fluctuations arise if the observable measured is such, that the state of the system does not have a definite value of that observable (in mathematical terms it is not an eigenstate of the operator representing the observable).
A: The best way to get a handle on odd quantum effects is to look at what the effect would mean physically.
One prediction made from the idea of vacuum fluctuation is that a strong enough electric field should polarize those fluctuations. In this case we are talking about polarizing virtual electron-positron pairs.  This effect is called vacuum polarization. 
Another implication is the idea that two conducting plates placed close enought together should exclude some of the quantum fluctuations.  In this case we are talking about virtual photons being restricted.  This is called the Casimir Effect
One of the oddities of quantum mechanics is that (in a sense) the possibility of something happening can have an influence on what actually does happen.  Feynman has a great description of this idea called the path integral formulation of quantum mechanics.  
I'm a big fan of the popular level explanation of the path integral approach in his book QED: A strange theory of light and matter.
A: Just to offer a different interpretation, from the perspective of the path integral it is very natural to speak of vacuum fluctuations. To be rigorous, lattice field theories discretise spacetime so that the path integral is well-defined. We have ample evidence that lattice field theories (in particular, lattice QCD) are a correct way to formalise nonperturbative aspects of quantum field theories, for instance see a plot from this (now old) paper

where we can see that lattice QCD correctly predicts the light spectrum of QCD. Now, measurements of operators in the vacuum (for instance, the chiral condensate $\langle \Omega | \overline{q} q | \Omega \rangle$ are in principle carried out by numerically performing integrals such as:
$$\langle \Omega | \overline{q} q(x)  | \Omega \rangle = \frac{1}{Z} \int_\text{lattice} dU d\overline{q} dq e^{-S_E[U,\overline{q},q]} \overline{q} q(x)$$
where the lattice integral is a finite dimensional integral. Here, I am ignoring all technical issues about renormalization and such. You will often hear people refer to this path integral as representing vacuum fluctuations, for example take a look at this gif from the Adelaide group showing typical gauge configurations that appear in the lattice path integral I wrote above

Each frame of the above gif would correspond to a single gauge configuration appearing in the path integral, on which you would measure some observable (such as $\overline{q} q$). Another way to think about it is that instantons (which are spatially extended, topologically nontrivial gauge configurations) cannot show up in perturbation theory, and are expected to contribute proportional to $e^{-1/g^2}$ to physical quantities, but on the lattice we do see instanton fluctuations, where they show up in our path integral measure. This is crucial to the understanding of nonperturbative QCD.
Addendum : Also just to mention, you can on finite lattices (in principle) compare the interacting vacuum state $| \Omega \rangle$ to the free vacuum state $|0 \rangle$ (the one which we would naively say contains `nothing'), and we see that they are different. In fact, Haag's theorem (or at least a part of it) tells us that their overlap goes to zero as you take the infinite volume and continuum limits. I think it's a fine intuition to think about the interacting vacuum as containing fluctuations, so long as you can be precise about what you mean.
A: Vacuum fluctuations exist, but they are not happening. The whole popular imagery surrounding the notion of vacuum fluctuations (and the associated virtual particles) is completely unsupported by the mathematics behind quantum field theory. It is solely created for the purpose of illustrating abstract concepts for an audience that likes imagery and mystery but has no understanding of the substance of quantum mechanics. Taking this imagery seriously leads to a host of unsurmountable difficulties. See my essay 
''The Vacuum Fluctuation Myth''.
A: Most of the previous answers argue that there are no actual quantum fluctuations. Nevertheless, we observe the effect of the quantum fluctuations of fields in the cosmic microwave background and in large scale structures (cosmic web).

According to modern cosmological theories, the quantum fluctuations of fields serve as seeds for the current inhomogeneities in the universe. Due to the very fast expansion during the inflation we can think of these inhomogeneities as a snapshot of the field values during that time. 
Of course, we don't really know what happened 15 billion years ago, but if we trust our models, actual quantum fluctuation in the field values are necessary to describe the observed universe.
For more details you can look into this lecture notes or the book of Prof. Mukhanov.
A: It is not needed for the vacuum to fluctuate all the time, but one can say the the probability of having a vacuum fluctuation at this point $x,t$ is non-zero
A: I think the misunderstanding is nomenclature. There is a physical effect on things in "empty" space I. E. Casmir effect... Some sort of fluctuating sub microscopic field exists. It's been measured, and some, including me believe that the reality of matter (energy, information) is some kind of manifestation of this field or aether. Recent discovery of frame dragging, among other things, shows some sort of field that somehow manifests everything. Call it what u will, but obviously the universe doesn't just chugg along without some sort of input energy or sea of potential energy that the physical world derives itself from. Your right when u say u cant get something for nothing, so something has to power the forces of nature, the manifestation of matter and energy, it just doesn't magically happen because close minded physic professors don't want to have to buy all new books...
A: It's not a finite event, there aren't individual fluctuations, there are infinity of them, that can't be observed, only the effects can. When there's nothing to fluctuate, no time nor space, then there's the big bang. Essentially the energy fluctuations were divided by nothing.
I would expect the opposite equation to have the exact opposite results. While infinite energy fluctuations spending 0 time outside of spacetime, if multiplying by infinite could have the opposite results as diving by 0, then instead of a material universe, entirely mathematically, bound to the laws of physics, which seems to prevent consciousness and purpose, we'd have an existence that's conscious, non-physical, bound by the opposite of our laws of physics. Second law of thermodynamics in reverse seems to define purpose, and with a physical brain only concerned with balancing emotions, having the opposite laws competing against that would be like creating free will.
However, we have to accept infinite fluctuations * 0 = nothing because it would be unscientific to consider anything else. That would make consciousness and purpose nothing more than a purely mathematical equation, considering the only thing we are sure of that is completely random, are quantum fluctuations. Considering the total energy in the universe is precisely 0, does that not make our physical universe just a mathematics equation? Seems like we need something else that isn't mathematical for mathematics to exist as anything more than theory but only random energy fluctuations can be entirely random and not mathematically, without our laws of physics being violating. Is considering there may be a missing piece to all there is being purely mathematical, really unclear and unscientific?
Energy fluctuations could still be utilised for artificial intelligence though, if you don't want the Chinese room argument from applying, as it always does with just code alone. The neural network could be still be produced with code, but instead of reading the values of the nodes in code and relaying on an algorithm to mutate the values to simulate evolution, or relying on a back propagation algorithm to reduce the margin of error, the nodes could be read from an em field. Provided outside interference is blocked by a faraday cage, then the driving force could potentially be quantum fluctuations, instead of an algorithm mutating values. Thoughts that need greater focus could be contained in the field, while other simultaneous processes could just use regular simulated neural networks in code alone. When nodes are close to firing but can't quite fire, energy fluctuations could trigger them sometimes. Identical code would no longer produce identical results. Using a different random seed for an algorithm based on time, is never the same as true randomness. Good enough for online casinos, but too fake to consider evolution to be genuine and not just a simulation.
A: For the ground state of  a mode of certain frequency omega, there are actually continuously  infinite number possible modes of plane waves each with a k vektor pointing towards a different direction in space. It is not true that phase doesn't matter. Yes it is true that the absolute value of the phase has no physical meaning but the relative phase between two waves matter. Since the relative phases of continuously infinite number of waves are random. The net result is a fluctuating vacuum field.
There is also an old myth telling "a photon interferes only with itself". It has been long time since the observed quantum beats caused by interference of the emitted radiation with two different frequencies in three level atoms debunked this myth.
