What are the dimensions, width and length, of a photon? Everyone is always talking about photon's wavelength. But what about its dimensions?
What is length and width of it?
And does it even have a point to think about such things? Or those dimensions are non-existent in such cases?
 A: The fundamental particles we know today (of which the photon is one) are called fundamental exactly because they have no substructure, or indeed, spatial extent, we know of. They are point-like when localized.
Note that these "particles" are quantum objects, not classical particles, so you should not imagine them as points whizzing about in space - they possess delocalized states where they take no definite shape at all (for example, the "electron cloud" around atoms is such a delocalized state).

The above is a short, non-relativistic view of "particles". When going to the relativistic description that is actually needed for the full description of fundamental particles, things get considerably more murky. For one, we lose the naive position operators, and the notion of "localization" becomes a bit ill-defined because the new "position operator", the Newton-Wigner operators, do not allow to speak of localization in an observer-independent way. The generic particle state that is scattered in QFT calculations is usually a sharp momentum state, and therefore strongly delocalized, so any notion of "point-like" can't really rely on the localization of a particle state.
In this picture, the proper notion of a "point-like" particle is one whose scattering behaviour indicates no substructure or spatial extent. For extended objects consisting of subobjects, their scattering behaviour will typically change when the energies/length scales of the scattering process reach their size, because then their internals get resolved and the individual subobjects start participating in the scattering. So then our notion of size becomes that the scattering behaviour is scale-independent. For more on this notion of size in QFT, see e.g. this answer by Bosoneando.
A: For me I prefer an experimental definition of the size of a photon. If you pass light through an aperture you start to see interference effects when the aperture approaches the wavelength of the photon, as if you're clipping the edges. Why do we need to make it more complicated than this?
If the photon is point-like then the energy density would be infinite which seems unrealistic.
It must be localized in space as photons can be detected from the other end of the universe. If photons are spreading out then their energy density would tend to zero over these distances.
A: The photon can be experimentally shown to not be point-like. The Young's slits experiment involves the interference of a photon with itself (the photon behaves in some ways like a particle, in some ways like a wave and in some ways like a probability distribution. in reality these are convenient models that we apply to it - in reality it is none of these - it is a photon). The interference patterns shown in the Young's slit experiment remain even if the stream of photons is reduced to the extent that photons go through 1 at a time. The length difference in the 2 paths can be varied to determine the coherence length. I believe this turns out to be in the order of 1m.
Further experimental proof that the photon must have a significant length is that the spread in its frequency is minimal. if it reduced to 0 amplitude over (e.g.) only 3 wavelengths then this would give it a significant spread in frequencies
