There is good book "Tables of Integral Transforms" by Bateman & Erdélyi where a lot of commonly used Fourier integrals are collected.

Is there a similar comprehensive collection of 2D and 3D Fourier transforms of functions occurring in physics and mathematics?

Update:

Below I give a few examples of Fourier transforms, which (and similar to which) I was hoping to find.
$$
\frac1{\omega^2-k\_x^2-b^2},\;\;\;\;
\frac1{\omega^2-k\_x^2-k\_y^2-b^2}, \; b\in\mathbb{C}
$$
$$
\exp\left\{{\rm i} z \sqrt{\omega^2 - k\_x^2}\right\}, \;\;\;\;
\exp\left\{{\rm i} z \sqrt{\omega^2 - k\_x^2 - k\_y^2}\right\}
$$
The Fourier transform here is to be calculated with respect to $\omega$, $k\_x$ and $k\_y$.
All these integrals can be calculated and have quite simple closed form.