might've /ˈmaɪtəv/ is the contraction of might have. The 've spells the reduced /əv/ sound, not the word of. I might've left my keys at work.
Keep the two words tight when you say it: /ˈmaɪtəv/. Written out fully, it stays might have, two words and no apostrophe.
I might've missed the train, so I walked home instead. Here the contraction slots in just before a past participle, matching the full form might have.
You might've seen that film already. Said naturally, have shrinks to the weak /əv/ after the /t/, so the whole thing flows as a single spoken unit.
Spelling trap: 'might of'
The spelling trap is might of: it looks wrong on paper but sounds identical, because contracted have takes its weak form /əv/, the same unstressed vowel you hear at the start of about. Cambridge Dictionary displays pronunciations in slashes using IPA symbols, and /ˈmaɪtəv/ follows that convention. Your ears are right; only the spelling is at risk.
Quick check when writing: if you can expand it to might have and the sentence still works, write might've. If you truly mean the noun or preposition of, as in a cup of tea, then of is the correct word, never after might.
In speech and informal writing, might've is the usual choice. For formal writing, prefer the full form might have.