Solidity is an illusion.
You may or may not already know this. Matter is mostly empty space: when you smack your hand against a table, what prohibits the further movement of your hand is the interaction of electrons, protons, and neutrons. At base, everything is likely a point particle, and all appearance of volume is caused by energetic excitations.
When you fire one point particle at another point particle, from a strictly geometric standpoint, the probability of collision is 0%. Nothing should ever hit anything else. And yet, two electrons launched at one another will "bounce"; the reason there being the electromagnetic repulsion. To account for this discrepancy between the expected geometric probability of scattering and the empirical measured scattering caused by the interaction, physicists who study such collisions use a quantity called a scattering cross-section. A scattering cross section is, more formally, a fictitious area describing the strength of interaction between two particles. This is given as a ratio: number of scattered particles divided by total incoming particles.
This ratio can be measured empirically in the lab by mere bean counting, but it can also be derived theoretically from considerations of the interaction potential. This is how we know the majority of what we know about anything on scales smaller than molecular. The existence of the nucleus within the atom, for instance (as opposed to Thompson' plum-pudding model) was discovered through a scattering experiment. We only know about quarks and the strong interaction through scattering. The recently discovered Higgs particle is also a result of scattering experiments. In all of these cases, just bouncing particles off of something and measuring the exact way that the particles bounce is enough to tell us what a thing is made of, how it is shaped, and -- more importantly -- the kinds of interactions that it undergoes.
Visible light is not normally useful to this purpose at subatomic lengths, but actually normal vision is an example of a kind of scattering experiment. Light from a bulb bounces off of an object and to your eye: you in a sense "measure" the angular deflection and intensity of this incoming light, and can thus determine the size, shape, and color of the object in question.
All of the things that you can see scatter light because all of the things that you can see are made of charged particles. Charged particles participate in the electromagnetic interaction, as does light, which means that normal matter is able to scatter light (as opposed to, say, dark matter). Were it not for the interaction (or coupling) between light and matter, then the electromagnetic cross-section of matter would be zero; light would see every surface as having zero area and therefore not bounce off of it.
To make this point more clearly, consider the neutrino. Neutrinos are not known to participate in any interaction besides the weak interaction. Therefore, neutrinos can fly right through the planet without slowing down. They're not flying through it like bullets, boring tiny holes; they're just flying through it. The solid matter of the earth is, to them, intangible and ethereal. They don not undergo the electromagnetic interaction, and so do not "see" the earth there.
I say all of this as introduction. What I really want to discuss are angels. In particular, how do we see them?
There are a number of accounts of people seeing angelic beings -- in the Jewish Torah, in the Christian New Testament, in the Islamic Quran, in reported private experiences, and likely in other religious texts that, sadly, I haven't studied. In some of these sightings, the angelic beings touch objects or people, and are able to move objects through (apparently) this touch.
Whether any of these sightings were real or not is a totally uninteresting question at the moment. You may, with rationalist skepticism, assume the accounts to be fabrications or psychological experiences; or you may, with faith, assume the accounts are real; or you may accept only those that agree with your religious views; you may also, in place of angels, substitute ghosts, or spirits, or gods, or some other non-physical entity of your choosing. None of any that will affect the point that I am about to make.
If angelic beings exist and can truly be seen by mortal men, and the sightings are not limited to the viewer's mind or imagination, then there are only a few options. Either an angel scatters light, or an angel scatters or emits some new particle not previously known that is able to interact with the photodetectors in the human eye in just such a way as to lead to normal vision. The latter would be very interesting, but already present too many unknowns to really consider in depth; therefore, let's consider the former case, that angels can scatter light.
How do they do this?
Again, there are a few options. The most obvious explanation of why a thing scatters light is that it is made of charged matter: protons and electrons. These form in to atoms, which form in to molecules; then there would be some matter distribution that corresponds to an angelic being. While this is physically possible, it is unsatisfying as an explanation. In all traditions, angels are pure spirit; it's hard to call something pure spirit if it is made of arrangements of carbon and oxygen. Therefore, I'm going to consider the case that more closely accounts with how angels are proposed to exist; angels are not made of matter, charged or otherwise. (I don't pretend to have proven this about angels, btw, I'm merely assuming this option for the sake of argument).
If angels are to be taken as physically visible (and not mentally or spiritual visible, as in visions or dreams) and if angels are to be taken as immaterial and not made of matter, and if the means of seeing angels is the normal means of human eyesight, then it must follow that angels are in possession of some hitherto unknown mechanism of coupling to light. There must be some new interaction that takes place within angelic beings that causes light to scatter from them. This interaction is not electromagnetic, because that arises from the interaction of charge. While it is unclear what this new interaction might be, under the assumptions given, it has to be there.
Now, believers in angels will likely object at this point (if they haven't already), that I am going way too far with this. Angels aren't physical things, after all, so ascribing physical properties like coupling and interaction potentials to them is silly. They're visible because they just are, that's it, no physics about it.
Except, supposing that I were to have an angel willing to present itself for study, I could place it in a ring of photodetectors, shine a beam of light on it, measure the distribution of reflected photons, and thence deduce both a scattering cross-section for my angelic visitor as well as an effective interaction potential between the angel and light. Whether a physical potential is the means by which an angel scatters light or not, I would still measure a physical potential anyway. I'd have to.
Even granting some spiritual, other-worldly basis by which angels scatter light -- one with no grounding in either physics or the natural world -- just doing the bean counting and then some algebra, a physical interaction is still what I measure. Call this an "effective interaction" if you will, and think of it as the physical imprint of whatever the spiritual reality may be. This effective interaction still scatters light, and therefore leads to a scattering cross-section. If there's a scattering cross-section, then there's an interaction potential. This potential could then be described, at least approximately and to within error bars, by algebraic formulae.
We'd then know some new force of purely spiritual entities upon physical matter. All we're lacking is an angel willing to participate.
While the experiment is impossible to perform (no angelic volunteers), and extremely ridiculous (measuring the properties of angels? really?), I think there's a lot of interesting points that come out of this sort of consideration.
One is a more clear understanding of how anything is visible at all: light collides with charged particles in matter and scatters. The charge represents the coupling strength of electromagnetic fields to matter, and therefore represents the interaction between them. Further, it is only this interaction of charge with light that causes the scattering, not any kind of "solidness" of the objects in question.
Two is a more clear understanding of how everything we do day-to-day works, down at the bottom of it. There are points, and the points have interactions, and these interactions are what cause everything that happens to happen. Sometimes you hear in sci-fi about aliens made of pure energy: well, so are you!
I think the main point, though, is that even in the most extreme case of a literal angelic visitor literally appearing, things can always be reduced to scientific and naturalistic explanations. Such is the nature of science.
For an angel, we can explain its visibility by measuring the effective interaction potential. This effect interaction may turn out to correspond to the potential of some particular arrangement of matter -- the angel absolutely is not made of matter, or of anything really, but scientific measurement of its scattering process will indicate matter regardless. Most scientists would then conclude (not unreasonably) that therefore that's what angels are made of.
If a person were to walk on water, hypothetically, a scientist making observation of this would conclude that some additional force were present under the foot, or that the surface tension of water was altered, or that the water molecules were coalesced under the foot to provide extra fluid pressure, all due to some observable process involving the acceleration of molecules. And where molecules accelerate, there is by definition a force.
If a person were to turn water in to wine, an observing scientist would notice the spontaneous creation of hydrocarbons and the vanishing of oxygen atoms. With enough measurement, the scientist could likely describe this process in terms of Feynman diagrams and particle creation/annihilation. If not Feynman diagrams, then something. An explanation can be found, and it can be found based on the observations of the event.
Whatever miracle occurs, there will always exist a naturalistic, scientific explanation by virtue of the fact that the movement of matter and light can always be described in terms of forces, potentials, and interactions, by the very definition of these concepts. Natural descriptions will always exist, and these descriptions will always be taken to suffice.
If nothing else, this makes the philosophical proposal of naturalists and materialists ("All you touch and all you see is all you are and all you'll be") an unfalsifiable claim. If a hypothetical, literal angel from heaven (were such to ever exist) can be rendered in to terms of natural scattering and interactions, then it is no longer a meaningful scientific question if there exists anything beyond the natural world. It is definitely not the final and certain conclusion of scientific progress.
At the same time, it renders all claims of supernaturalists scientifically unfalsifiable. If I can show that your so-called angel behaves no differently in the presence of light than does normal matter and therefore must be made of matter, then you can rejoin that that is only what my measurements say, contrary to reality. After all, as I explained above, even a completely ethereal, spiritual, non-physical angel, were it capable of being detected at all, must be describable in terms of effective interactions. No experiment will be able to discern the difference. Therefore we are also beyond falsifiability, and the nature of your angel is no longer a meaningful scientific question.
Science will always say that every phenomenon is natural, no matter what it is, because science will always only measure its natural properties, and will always be able to describe those natural properties in natural terms.
It is impossible to distinguish between a strictly natural, rationalist universe with nothing but the purely physical laws, and a universe full of supernatural elements and miracles.
Also, for those of us who do believe in literal angels that are made of pure spirit and yet have, at certain times, shown themselves to human people, there is an interesting quandary here. What is this effective interaction? Whence does it arise? How do angels apparently turn it on and off at will? What implications are there for this force capable of coupling to both light and matter? What models might describe it? What does this say about the nature of angels and other heavenly beings?
Really, there's no way to answer any of this, because angels, even if they existed, would not willingly subject themselves to scientific experiments for our amusement. I just thought it was fun to think about.