I'm not a teacher, but I would say that "on purpose" is the most casual of the three. It's the phrase one would expect to hear from a child complaining of some injustice - "You did that on purpose".
In everyday informal settings, I'd be most likely to hear and use "deliberately". My wife will often say "that car deliberately sped up as soon as I had a chance to pass, then slowed down again". In a sentence like that one, "intentionally" would sound a tiny bit unusual, to my ears. I use "intentionally" most when writing. For example, someone once thought I had written "contemn" instead of "condemn". I replied that I had intentionally used "contemn", because it was my intent, it was the word I had intended to use.