As far as I knew*, all English syllables have a vowel sound and all of them are spelled accordingly, except for “thm” as in rhythm and algorithm. Are there any others? And are there any etymological reasons why this / they exist(s)?
* See JSBang’s answer.
Occasionally -sm does the same thing: chasm, schism, etc. As I pronounce them, these are all two-syllable words.
Having said that, I would question your premise that "all English syllables have a vowel sound". There are in fact a great many English syllables which don’t have any vowel sound at all (in most US English dialects, as discussed below), but rather have a syllabic consonant:
The second syllable of all of these words, though spelled with a vowel, is typically pronounced with no vowel sound at all between the medial consonant and the final consonant. Instead, the final consonant is elongated into a syllable of its own. In pickle, for example, there is no vowel, not even a schwa, between the [k] and the [l]. As soon as the [k] is released the lateral contact on the [l] begins, and the [l] sound is drawn out for the full length of an unstressed syllable. In my dialect, at least, all words ending with an unstressed syllable containing [n], [r], or [l] are pronounced this way.
Different dialects handle this differently, however. In British English, for example, tanner often has a final shwa and no [r] sound at all, and the handling of unstressed final [n] as in button varies quite a bit even within North America.