Do physical entities with one dimension equal to 0 exist? Within mathematics, a lot of entities with one or more size equal to zero exist and are fluently manipulated.
For example, within $\mathbb R^4$, the entity defined as $z=0, t=2$ is a plan, the one defined as $x=ct, y=0, z=0$ is a line, and the one defined as $x=0, y=0, z=0, t=0$ is a point.
Do this mathematical entities have a real counterpart?
For example does there exist any physical entity which one spatial dimension would equal 0? By 0 I mean here a real zero (not $0 \in \mathbb R$, and not $0±10^{-15}$) which might be unreachable to us due to the Heisenberg's uncertainty principle, but nonetheless physical and even provable due to its consequences.
For example 2 such objects with let's say $\delta x = 0$, could occupy the same spatial position $x=x_0$ at the exact same time $t=t_0$.
As a typical example, the electron is said to be a "point like" entity. This expression seems to have been carefully chosen to avoid as a bad word the expression of a "null size" entity.
My naive point of view is that null size "can't exist" because it leads to:


*

*infinite physical densities,

*impossibility to disambiguate 2 such null entities (2 particles of null size, with all other physical characteristics identical, can occupy the same position at the same time, and nothing can help to name one out of the 2),

*infinite number of such entities in the exact same position.


Is the existence of such entities proven within the field of elementary particles or electromagnetic waves?
 A: I'm going to say no, because I interpret a "physical entity" as something which we can observe, and therefore confirm it's existence. For instance:
Particles (0 dimensional): Mathematically are points, but when we observe them we observe them to have sizes because of the observation process (bouncing photons off atoms, electrons off other electrons, even at the level of fundamental particles we have the quark cloud).
Strings (1 dimensional): In principle, strings have only one dimension, and zero in the others. However, we haven't observed these kinds of strings (aka String Theory or Cosmic strings). Strings which have been observed (in condensed matter systems, I think) have finite size because of the observation process I discussed above. Loops in quantum gravity go here, but similarly have not been directly observed.
Surfaces (2 dimensions): "surface of last scattering" from the big bang is fuzzy because of particle excitations. Maybe the event horizon of a black hole is a good example, but most people now think that this is also fuzzy once you look closely enough - and we still haven't directly detected one.
Essentially, nothing which is made of matter can have zero spatial or time extent because of the observation process. 
A: Do any physical real entities exist?
All the particles we consider elementary (the most well-known being the electron) do not occupy any finite amount of space, i.e. are pointlike, at least according to our present understanding of the standard model. However, that understanding could change as new theories emerge or as experimental evidence is found.
As for two particles occupying the same point at the same time, the Pauli principle still applies, i.e. two indistinguishable fermions (such as the electron) can not occupy the same state (as characterized by the position) at once, while bosons on the other hand do not care.
Now, for the matter of real physical existence, quantum field theory describes particles as excitations of a corresponding underlying field that permeates all space(-time). It is up to you if you want to believe that the particle is real or the field, or... anything? String theory introduces a bunch of other objects that would match your description. It is not up to physics to answer the ontological aspect of your question.
A: 
My naive point of view is that null size "can't exist" because it leads to:
infinite physical densities,

In classical physics; the underlying level of nature is quantum mechanical , as far as our experiments and the theory that describes them. There are no infinite densities.

impossibility to disambiguate 2 such null entities (2 particles of null size, with all other physical characteristics identical, can occupy the same position at the same time, and nothing can help to name one out of the 2),

It would be true classically but  not quantum mechanically,  because there exist quantum numbers that are conserved and separate particles . Elementary particles are point like,  quantum mechanical entities whose interactions can be predicted using the mathematics of Feynman diagrams. For example in Compton scattering

A photon and an electron meet at a mathematical point , an integral over the variables will give the probability density for the outgoing photon and electron. The whole process in real space is bounded by the Heisenberg uncertainty, which is inherent in the quantum mechanical mathematical model. Thus, the model has point particles and overlaps, the measurements have an inherent quantum mechanical uncertainty.

infinite number of such entities in the exact same position.

The very successful quantum mechanical theory, gives point particles which have  3 point vertices only, so no infinities . One has to understand that one has to develop a new intuition for quantum mechanics. One cannot extrapolate classical physics behavior to the elementary particles.
All macroscopic physical entities are constructed by elementary particles, and their dimensions are limited by the Heisenberg uncertainty and the statistical distributions of many bodies, so until one reaches the elementary particle level there are always three dimensions to them. At the elementary particle level the answer is yes, at the theoretical level, bounded by the HUP at measurement level.
