What we observe in nature exists in several scales. From the distances of stars and galaxies and clusters of galaxies to the sizes of atoms and elementary particles.
Now we have to define "observe".
Observing in human size scale means what our ears hear, what our eyes see, what our hands feel, our nose smells , our mouth tastes. That was the first classification and the level of "proxy", i.e. intermediate between fact and our understanding and classification, which is biological. (the term proxy is widely used in climate researches)
A second level of observing comes when we use proxies, like meters, thermometers, telescopes and microscopes etc. which register on our biological proxies and we accumulate knowledge. At this level we can overcome the limits of the human scale and find and study the enormous scales of the galaxies and the tiny scales of the bacteria and microbes. A level of microns and millimeters. We observe waves in liquids with such size wavelengths
Visible light is of the order of Angstroms, $10^{-10}$ meters. As science progressed the idea of light being corpuscles ( Newton) became overcome by the observation of interference phenomena which definitely said "waves".
Then came the quantum revolution, the photoelectric effect (Particle), the double slit experiments( wave) that showed light had aspects of a corpuscle and aspects of a wave. We our now in a final level of use of proxy, called mathematics
The wave particle duality was understood in the theory of quantum mechanics. In this theory depending on the observation a particle will either react as a "particle" i.e. have a momentum and location defined , or as a wave, i.e. have a frequency/wavelength and geometry defining its presence BUT, and it is a huge but, this wavelength is not in the matter/energy itself that is defining the particle , but in the probability of finding that particle in a specific (x,y,z,t) location. If there is no experiment looking for the particle at specific locations its form is unknown and bounded by the Heisenberg Uncertainty Principle.
What is described with words in the last paragraph is rigorously set out in mathematical equations and it is not possible to understand really what is going on if one does not acquire the mathematical tools, as a native on a primitive island could not understand airplanes. Mathematics is the ultimate proxy for understanding quantum phenomena.
Now light is special in the sense that collectively it displays the wave properties macroscopically, and the specialness comes from the Maxwell Equations which work as well in both systems, the classical and the quantum mechanical, but this also needs mathematics to be comprehended.
So a visualization is misleading in the sense that the mathematical wave function coming from the quantum mechanical equations is like a "statistical" tool whose square gives us the probability of observing the particle at (x,y,z,t). Suppose that I have a statistical probability function for you, that you may be in New York on 17/10/2012 and probabilities spread all over the east coast of the US. Does that mean that you are nowhere? does that mean that you are everywhere? Equally with the photons and the elementary particles. It is just a mathematical probability coming out of the inherent quantum mechanical nature of the cosmos.
Different possible polarizations of a "matter particle wave" corresponds to the different possible degrees of freedom of the quantum field describing the "particle".
For a photon, we have 2 possible polarizations (for instance : vertical polarization, horizontal polarization). For a electron, we have also 2 possible polarizations (for instance : left handed, right handed). For the positron, we have also the same 2 possible polarizations , and the whole electron/positron quantum Dirac field describes 4 possible polarizations.
However, transversality has to do with a precise space-time condition, and this notion is only available for some Lorentz representations. A transverse relation will be written : $\vec k.\vec \epsilon_\lambda (k) = 0$. However, it suppose that the Lorentz representation of the field is a "vector", which is (roughly) true for the photon field, but false for the electron/positron Dirac field. In the latter case, the representation is a bi-spinor, so you cannot get a transversality relation directly between the momentum $\vec k$ and a bi-spinor like $u(\lambda, \vec k), v(\lambda, \vec k)$ (you will have to involve bilinear (quadratic) quantities based on bi-spinors to get "vectors").
In the same way, the notion of longitudinal wave $\vec k$ parrallel to $\vec \epsilon_\lambda (k)$, is a nonsense in the case of the Dirac field.
Best Answer
A deBroglie wave has two interpretations, which are generalizations in different domains, and which are conflated for a single particle. This leads to a lot of confusion.
historically, Schrodinger interpreted the deBroglie wave as the first thing initially, as a physical scalar wave. This is the wrong interpretation, as it is not equivalent to matrix mechanics, and it is experimentally untenable since a physical wave doesn't allow for entanglement. The battle over this was settled by Schrodinger (and Einstein and deBroglie too, who understood the deBroglie wave was like the solution to the Hamilton Jacobi equation, something that lives in configuration space), who demonstrated that the wave was in configuration space in 1926, and proved that with this interpretation, the Heisenberg formalism was a consequence of the wave formalism.
To quickly answer the questions
The points above require a little more discussion, regarding the field and particle intepretation.
When deBroglie understood the matter waves, it wasn't clear if these are physical waves in space, like an electromagnetic wave, or if they are something more abstract, like the solution to the Hamilton Jacobi equation. The Hamilton Jacobi solution is over all classical configurations, and it tells you what the integrable motion frequencies are. Einstein established the character of the deBroglie waves in 1924, by showing that the semiclassical limit description, they are the solution to the Hamilton Jacobi equation. When Schrodinger found the right equation, Einstein and Schrodinger discussed the interpretation, and it became clear that the Schrodinger equation too was to be thought of as a wave over configurations.
What this means is that the wave for 2 electrons is in 6 dimensions, for 3 electrons in 9 dimensions, describing all possible mutual positions of these. This led Einstein to ask how physical these waves are, considering that if you have a powderkeg in quantum mechanics, you can set up a situation where its wave is superposed between exploded and unexploded. This observation of Einstein's is the origin of Schrodinger's cat, and it is the reason Einstein could never be convinced to take the quantum formalism seriously as a description of physical reality--- it was just too enormous to be physical. It looked like a statistical description of something else. This has not been a common interpretation, because if it is a statistical description of something else underneath, we don't know exactly what that other thing could be.
But before chatting with Einstein, Schrodinger believed his equation described ordinary scalar waves in space. This interpretation made the amplitude $|\psi|^2$ a charge density, and the Schrodinger current an actual electromagnetic current.
While this interpretation is incorrect for the fundamental quantum deBroglie wave, it is correct for a Bose Einstein condensate. If you have many Bosons in a superposition state where they all share the same quantum state, their wavefunction becomes a classical field which obeys the Schrodinger equation, a Schrodinger field. The Schrodinger field description does not require linearity, it is just a scalar field (or a vector/tensor field for bosons with spin) which describes the density and matter-current in a Bose Einstein condensate. In this context, it is called a Gross-Pitaevsky equation, or in other contexts, a Bogoliubov-deGennes equation, or something else, but this field interpretation is very important, because it is the only limit in which Schrodinger waves turn into waves in space.
In this context, the deBroglie wave shared by the Bose particles turns into a classical scalar field, and it has an intepretation which is identical to the one proposed by Schrodinger. But such a description cannot describe entanglements in nature, and the simplest case where entanglement is seen to be necessary is in the ground state of the Helium atom.