I think this is a trivial matter in the sense that no one will fault you if you spell it bi-monthly or bimonthly. Anyway, the entry is under bimonthly at dictionary.com and Merriam-Webster bimonthly. The examples given at dictionary.com are from works that are at least one hundred years old. If we look at some search results, we see that bimonthly is more popular.
There's not very many patterns that would help in most cases. Small patterns exist but you are better off simply memorizing the spelling of common -ce and -se words.
For example - the c in -ace or -ice at the end of words is almost always pronounced /s/, whereas the s in -ase or -ise would be usually pronounced /z/. So use -ce in that situation.
There are far fewer words that end in -ense than -ence, and likewise less that end in -se with s pronounced /s/, rather than -ce. So if you have to hazard a guess, go with -ce.
The Punctuation Guide is a good source of information for English punctuation and its usage, including what each symbol is called (though in this case it doesn't mention the tilde).
Typing the actual symbol into Google or Wikipedia also yields an accurate result.
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If you type a punctuation symbol into Wikipedia (such as tilde, above) it also returns a comprehensive list of other punctuation symbols, spellings, plus links to more detailed information on the right hand side of the page.
Best Answer
I think this is a trivial matter in the sense that no one will fault you if you spell it bi-monthly or bimonthly. Anyway, the entry is under bimonthly at dictionary.com and Merriam-Webster bimonthly. The examples given at dictionary.com are from works that are at least one hundred years old. If we look at some search results, we see that bimonthly is more popular.
So all in all, I would write bimonthly.