What is the correct definition and usage of "for all I know"?

I saw some sentences using this phrase "for all I know" but wasn't quite sure what is its exact definition and also whether a modal verb such as "could, may or might" expressing uncertainty must be present in the sentence.

The following are some possible definitions:

• According to the information I have; I think; probably. (usually implies uncertainty)
• Based on what I know (usually expressing displeasure)
• As far as I know (used to show indifference)
• I really don't know (used to show complete ignorance)

And these are statements containing the phrase:

• For all I know, the mayor has resigned already.
• She may have gone to town for all I know.
• He wears a ring, but he may be single, for all I know.
• For all I know, there may be someone with you now.
• The man I met on the beach yesterday seemed friendly, but he could be a thief, for all I know.
• For all I know, the girl was buried alive in the Arabian sands.
• They've decided to hire Jack for all I know.

References:
http://idioms.thefreedictionary.com/for+all+I+know
http://forum.wordreference.com/showthread.php?t=22340
http://esl.about.com/od/vocabularyreference/a/all_expresions.htm
http://dictionary.reverso.net/english-definition/for%20all%20i%20know
http://dictionary.reference.com/browse/for+all+i+know
http://www.classic-country-song-lyrics.com/foralliknowlyricschords.html

What does the phrase "for all I know" mean exactly?

“For all I know” asserts that the speaker is so insufficiently informed that he is willing to entertain the most far-fetched conjecture or the darkest suspicion. In speech there’s usually one strongly stressed word or phrase to name that implausibility or fear.

And it almost always implies some measure of annoyance, which may be occasioned by any number of things: the reprehensible conjecture under consideration, the speaker’s own lamentable ignorance, the people who are maliciously keeping him in ignorance, being questioned on a subject he doesn’t want to discuss.

There's really no way of telling what your specific sentences mean without a lot more context, but here are some guesses:

• The mayor may resign tomorrow, he may resign next week — for all I know, he’s resigned already. Who cares? He’s gone.
• She may be at her mother’s, she may be at the school, she may have gone to town for all I know. Get off my case.
• He wears a ring, so he may be married, he may be widowed, he may be single, for all I know. Or gay.
• You tell me you’re working, you tell me you’re alone, you tell me you miss me, but how can I tell? For all I know, there may be someone with you now, this very minute. You rotten SOB.
• The man I met on the beach yesterday seemed friendly, he looked honest, but he could be a thief, for all I know. You can’t trust anybody.
• The girl may have moved. She may have gone off to college. She may have entered a convent. For all I know, the girl was buried alive in the Arabian sands. All I know is, she’s not here.
• They’ve decided to hire me. They’ve decided to hire you. They’ve decided to hire Ben. They’ve decided to hire Britney. They've decided to hire Jack the f—ng Janitor for all I know. Nobody tells me anything.

For all I know is used when the speaker wishes to make a point that they don't know much at all, so they attach it to a statement that is either preposterous or undesirable from the speaker's perspective.

He wears a ring, but he may be single, for all I know. means that given what little I know if this person (which in this case is merely that he wears a ring) I don't know enough else to know whether the ring is for real or not.

For all I know, there may be someone with you now. sounds like someone's wife on the phone with their husband who suspects they've been cheating. The wife can only go by the denial their husband has just uttered, but has no real way of knowing whether it's the truth.

For all I know, the girl was buried alive in the Arabian sands. means that given what little I know being buried in the Arabian sands fits with the facts however preposterous it might be. Making the point that the speaker knows very little indeed.

For all I know is used to emphasise that you do not know something, while the similar for all I care is used to say that you are not worried or affected by something.