Probably the most often-heard example of hypercorrection in spoken English is one that I'm sure we have all made, from time to time: the use of 'I' as the object of a verb ("Let's keep this between you and I."). It's perhaps somewhat more common to speak of oneself in the active voice, so we are more likely to be berated by our schoolteachers for the opposite case, using 'me' as a subject ("Betsy and me went skinny dipping in the lake last night"). We over-correct to avoid making the 'mistake' of saying 'me,' and instead misuse 'I.'
Similarly, and an equally clear-cut example of hypercorrection, we have the use of 'whom' as a subject pronoun. Those of us who make many business calls every day probably hear this more often than we'd like. A secretary answers the phone, asks how she should direct your call, and then says, "Whom may I say is calling?" As in the first case, this is an overcorrection to avoid using 'who' as an object pronoun; the problem, of course, is that most people don't know when one or the other is required. They know that this word 'whom' exists, and that it somehow relates to 'who,' but they don't quite know how.
Hypercorrection is almost always the result of prescriptive grammar. Speakers are understood perfectly well when they say, "She's the dame who I like best," but are perhaps told that 'whom' is correct in this case ('the dame' being the object of 'I like'). Unfortunately, since they likely don't understand the reason for this -- understandably, since I've yet to hear of a public school that teaches such things -- they are likely to hypercorrect in the future, in order to sound more erudite. So please, do your part to stem hypercorrection; if you're going to correct someone's grammar, either explain the reason, or don't correct. I, for one, would rather have an army of people telling me what 'Petey and me' did yesterday than telling me how Betsy went down on 'Petey and I' behind the barn.