Gravitational waves are transverse waves but they are not dipole transverse waves like most electromagnetic waves, they are quadrupole waves. They simultaneously squeeze and stretch matter in two perpendicular directions. Gravitational waves definitely propagate in a given direction but the effect that they have on matter is completely perpendicular to the direction of motion. Below is a picture of what the metric of a passing wave does to space (the wave traveling is perpendicular to the screen). If you imagine a free particle sitting at each grid intersection point, the particle would move sinusoidally right along with the grid:
This diagram is from this paper
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
Sound is a pressure wave, alternating deviations of pressure from the equilibrium. So, depending on the medium in which the pressure wave passes, you can get either type of wave (longitudinal or transverse):
In gases and liquids, the pressure deviations causes compressions and rarefactions, meaning these are longitudinal waves.
In solids, the pressure deviations cause shear stresses along the perpendicular direction to the direction of motion of the wave, meaning these are transverse waves.
As far as I know, tuning forks are used in air, meaning they generate longitudinal sound waves.