Phoneme distribution

It is quite common to see tables of how different letters are distributed
in typical texts of a given language, but have there been any similar
investigations regarding phonemes of the spoken language?

There are a few things I am particularly curious about, such as whether
the various phonemes of a language typically follow some sort of pattern.
If, for instance, any given phoneme typically occurs in between 1% and 45%
of the words of a language (this example is, of course, total nonsense).
Or if diphtongs (when a language has such) are usually much less frequent
than single vowels, etc. (Hmm... perhaps a diphtong would count as a
morpheme rather than a phoneme? Oh, well, you know what I mean.)

I hope I am getting through clearly here. I do not feel extremely coherent
right now.

  /F