We never use more than one apostrophe to a word.
While the general rule is to use the apostrophe in place of the last missing letter, such as in "shall not -> shan't", if we need to choose between missing letters that we'd normally pronounce and those that are silent, use the apostrophe to denote the missing sounds.
Well. at first glance, it appears to me that our way of ‘making’ contractions is to 1) lop off the last half of the first word and 2) smash it together with the “not”, contracting the “o” with an apostrophe. See for yourself…
shan’t= shall (minus the “-ll”) + not (minus the “o”) = sha n’t
Which is then moved together (sha->n’t) to spell: shan’t. Personally, I believe that this contraction (judging by the way we use the word, and say it) is an ‘evolved creature’ from the two contractions “shouldn’t” and “can’t”; as opposed to its parentage being shall and not. It just makes more sense. [Shouldn’t + can’t= shan’t]
Did the contractions won’t and shan’t spring into English fully formed, like Athena from Zeus’s noggin? No, interestingly. Samuel Johnson’s Dictionary (printed in 1855), has wo’n't, as do some (modern) editions of Lewis Carroll’s Alice in Wonderland and The Ohio Educational Monthly in an article from 1868. Likewise, sha’n't was commonplace in the old days, plastered across the pages of the dreadful Victorian novels that I had to read in AP English as a lesson as to what happens to those who show an interest in reading. Books like Evelina; or, The history of a young lady’s entrance into the world (why did every single book in those days have to have a subtitle?)
Now the interesting thing is that won’t and shan’t live side-by-side with wo’n't and sha’n't in these old books. Some quick results on Google Books between 1600 and 1800: 777 won’ts, 57 wo’n'ts; 216 shan’ts, 73 sha’n'ts. Between 1600 and 1700: 48 won’ts, no wo’n'ts; 1 each of shan’t and sha’n't. So it seems it was never the case that the multiple-apostrophe form was more common. For some reason or another, English writers have always preferred a single apostrophe over strict application of “put apostrophes wherever a letter’s missing”. (Michael Quinion guesses that the double-apostrophe form was a later edition, suggested by logic-minded grammarians, that died out because it was a pain to write and looked weird.)