I'm interested in the frequency of pinyin syllables (both when including tones and toneless). This implies the usage in real text, weighted by the occurrences in the corpus, not by parsing a dictionary which would overblow some really rare syllables that occur in many bookish words. I might confidently assume that "de" of Netral Tone is the most frequent; but which one is the rarest (in everyday usage)?
Jun Da has a lot of frequency data on his website, including syllable frequencies both with and without considering tones, based on frequency data for the 3,500 most common characters. This is, in other words, indirectly based on corpus data, but does not include rare characters.
I'm not sure exactly why you're asking this question, so this may or may not be enough for you. Asking what's the rarest of anything is kind of pointless, in my opinion, as there probably are edge cases which might have an odd occurrence here and there. Where do you draw the line?
Any of the less common pinyin sounds would be rarest, ang2, chong3, or pie4 are possible contenders. looking at the pinyin without the tone is counterintuitive to me, since the two aspects are not separable in chinese. Of course, you could do it for fun, but it might be hard to get anything that makes sense as a result.
I think it is impossible to officially calculate the true rarest pinyin sound, since written text does not notate the pronounciation used for the characters. You would need someone to actually go through all chinese text and recordings to manually check what was used most, which is certainly not feasible.
However, if you look up rare pinyins that only belong to a few characters you can see the contenders, especially those that are literary chinese or otherwise uncommon. (for constrast ka is not that common of a pinyin, but the characters it is used for are very common-- things like cards and coffee).
的 is definitely the most common character written in modern chinese, and its de pronounciation the most common as well.
I didn't find a frequency table based on pinyin. However, I found one Chinese character frequency list 汉字字频表 based on Chinese characters. This table includes pinyin and frequency of each character listed in it. You may want to import the table to Excel and manipulate it to get what you want.