Are quantum fluctuations observable?

I believe there are cases where they are observable. Especially through their effects. In cosmology, for example, quantum fluctuations in the early inflationary epoch are considered to have seeded the large scale structure of the universe.

In a more 'accessible' example, quantum fluctuations are the cause of the Casimir effect, which can be measured. In this case, quantum fluctuations in the space between two metal plates, result in a net attractive force (which is quite small, orders of magnitude smaller than the gravitational attraction between them). There are more details in the wikipedia page https://en.wikipedia.org/wiki/Casimir_effect .

It has also been hypothesized that energy from quantum fluctuations in the vacuum contributes to the energy required for the accelerated expansion of the universe, even though there is an ~80 orders of magnitude discrepancy between what the vacuum gives and what the universe requires for its observed acceleration. It's one of the most important open questions in science.

If one of these cases interests you, perhaps we could look further into it.

Not only are they observable, they can be manipulated. Although different degrees of freedom are usually uncorrelated, nonlinear optics may be employed to create correlations that reduce electromagnetic quantum fluctuations in some variables at the expense of making them large in others. This trade-off is a consequence of the Uncertainty Principle. If you're making a measurement that's sensitive to to fluctuations in some variable while insensitive to fluctuations in its conjugate you may use this to improve your measurement. The gravitational-wave observatory LIGO pioneered this technology.

The Lamb Shift is pretty much considered to be the 1st observation of quantum fluctuations. It is a small energy difference between the $^2S_{frac 1 2}$ and $^2P_{frac 1 2}$ states in the hydrogen atom.

The relativistic theory of the electron, the Dirac equation, predicts that the electron binding energies in these two states have the same energy (so-called degenerate states) in the electrostatic field of a proton.

When the electric field is treated quantum mechanically, the 1st correction to the $1/r$ potential, which is mathematically modeled as the exchange of a virtual photon, has that photon quantum mechanically fluctuate into a virtual electron/positron pair. That pair then modifies the "static" potential near the origin (where the field is strongest) and "lifts the degeneracy", that is: it changes the energy of the S-state. (The S-state has strength at the origin, which colloquially is states as, "spends the most time near the origin"...but that's kind of a classical way of putting it).

So when we say we "see" the quantum fluctuations, we don't have a photograph or mp3 file of the field strength changing, rather we observe its effects on a measurable thing, here: microwave transitions in the hydrogen atom, and connect that with a theoretical calculation.