Distribution theory book

The book

Duistermaat J., Kolk J. Distributions: Theory and Applications (Birkhäuser 2010)

deserves to be mentioned.

Since I want to close this question, I will post an answer myself.

A Guide to Distribution Theory and Fourier Transforms - Strichartz is a nice introduction but it contains almost nothing.

Théorie des distributions by Laurent Schwartz is written by the master and father on the subject and therefore I say it is a good book.

Theory of Distributions by J. Ian Richards and Heekyung K. Youn is a self-described "non-technical introduction", which seems to mean you don't need to know functional analysis, measure theory, or topology. But you do need to think more like a mathematician than like a physicist or engineer; it's all mathematically rigorous. It contains the authors' original results on the question of when two distributions can be multiplied.

Distribution Theory and Transform Analysis by A. H. Zemanian develops the theory, then does Fourier and Laplace transforms, then applies it all to problems arising in engineering.

And there's Introduction to Fourier analysis and generalised functions by Sir James Lighthill.