Actually, I'm wondering if someone might come up with some weird grammatical explanation for "should of". :) We had this exact phrase in my class on modals this morning, and I explained "should of" for two reasons: 1) My students will encounter it sooner or later, and they should know it's a laughable mistake. 2) It explains how they need to pronounce "should have" if they want to sound natural. They were still pronouncing the correct phrase as two separate words, but "of" and " 've"(have) sound exactly the same in speech.

Bad English but acceptable American? : ) Or so it would seem, judging on the number of times I encounter this heresy on American websites. Who knows, it might soon be added to the Webster... I wouldn't bet against it!

No, it's not. It's a common error made by native speakers. Non-native speakers don't make this mistake because they learn modal verbs in a completely different way to natives. In speech, 'should have', is normally contracted to 'should've'. 'Should've' sounds like 'should of' when spoken. People write how they speak and we end up with the grammatical nonsense above. Also things like 'must of', 'will of', 'could of', you will see those too.

"Should of knew" is really, really bad English. There are two mistakes in it. This sentence ought to read "X should have known from the start". Writing 'of' instead of 'have' is a common mistake among many native speakers. In spoken English, the auxiliary 'have' is nearly always reduced to an 'uv' sound with a schwa vowel, so many native speakers presume that the word they're saying is 'of' rather than 'have'. The sound is correct, but the written form is wrong. Using a past simple (knew) instead of a past participle (known) is simply wrong. It shows a real lack of education. The kindest way you could describe this way of speaking is 'non-standard'.