I think 'persist' and 'persevere' can be used as absolute synonyms, that is to doggedly continue to do something:
If you persevere with your search for a job, you are sure to find it soon.
Even though everybody told him he was wrong, he persisted in his opinion.
Chances are slim for her to win this title, yet she still persists in trying.
'Insist' can have a similar meaning. For example, when someone stubbornly makes an allegation:
The prisoner insisted on his innocence.
However, I think more often 'insist' implies demand or even order. You're demanding something from someone when you insist on it:
I insist on the return of my money at once!
Anyway, all three words are pretty close in meanings, although there are nuances when each of them can have a slightly different undertone.