Your question is a little bit confused, so let me try to explain things slightly more systematically.
1. It is important to understand that there are actually many different types of waves, not three and not four. However, many waves are enabled by specific geometric features of the problem at hand, so one usually separates so-called body waves from all the other kinds of waves. Body waves can propagate in an infinite elastic space. There are exactly 2 types of body waves: longitudinal and shear waves. Generally speaking, in anisotropic materials, the velocity of these waves can depend on the direction of propagation and these waves themselves can couple to each other. However, assuming isotropic materials, you will have them uncoupled and their velocities will be direction-independent and given by $$ c_1^2=\frac{\lambda+2\mu}{\rho},\quad c_2^2=\frac{\mu}{\rho}, $$ where $c_1$ is the longitudinal wave speed and $c_2$ is the shear wave speed. You can see that the shear wave speed is proportional to the square root of the shear modulus. However, the velocity of longitudinal waves is not related to one specific modulus.
2. Any isotropic material can be characterized by exactly two modulae. 795
$\lambda$ and $\mu$ that I used just above are often called Lame parameters. In some situations people prefer to use other modulae too. Modulae most commonly used for isotropic materials are: $$ E=\frac{\mu(3\lambda+2\mu)}{\lambda+\mu},\quad \nu=\frac{\lambda}{2(\lambda+\mu)},\quad \kappa=\lambda+\frac{2}{3}\mu. $$ $\kappa$ is the bulk modulus, it characterizes stiffness for volume deformations; $E$ the Young modulus, it predicts the stiffness of a thin bar, and $\nu$ the Poisson ratio that characterizes how much thin bar contracts transversally when it is stretched.
3. Note the emphasis on specific geometries in the last two examples. Some combinations of modulae were introduced precisely because they are convenient to use for specific geometries. Since thin bars are convenient objects for testing, a number of parameters are defined with thin bar testing in mind. In this sense, you may find it instructive that velocity of sound in a thin elastic bar is given by $$ c_{\mathrm{bar}}^2=\frac{E}{\rho}, $$ i.e. that the Young modulus is directly proportional to the square of velocity of longitudinal waves in bars. However, outside of this specific geometry, the interpretation of Young modulus becomes much less intuitive. 333