1) "The thing is, is..."
Usage examples from Google:
a. "The thing is is that I do not want good looking girls to be around him..."
b. "The thing is, is that I might be moving in two years."
c. "The thing is, is that EVERY drug has some sort of side affect [sic]..."
d. "The thing is, is that they are back in school and there is an increase of children around the schools and on the streets."
Most editors would, wisely enough, point out that none of these sentences need the extra "is." "The thing is, I might be moving in two years." Perfectly good sentence. So why do so many various English speakers do this? As usual, LanguageLog has beat me to an examination. Their suggestion, which strikes me as pretty good, is that people are, in part, modeling such sentences after similar ones in which a wh- word is used. (Note: in the world of Ling, wh- words are who, what, when, where, how and their semantic equivalents, even if they don't actually start with "wh".) For, with wh- words, we get grammatical constructions like:
a. "What the question is, is what is that purpose?"
b. "What the question is, is not mentioned."
c. "What the issue is, is not Al-Qaeda but local war lords."
Wh- constructions are complicated and gave me headaches when it came time to draw trees for them in syntax class (do a Google search on wh- syntax if you doubt me), but the basic thing to notice is that when the subject of your sentence is a wh- clause, as in those above, you do still need a verb for the main sentence, and usually it is a form of "be." When you take the wh- word away, the sentence gets less complicated and you no longer have that subject clause, and thus you no longer need the extra verb. See?
The reinforcing reason why people might say "The thing is, is" is that humans tend to repeat words in speech, when stalling and thinking. Actual transcripts read something like, "I--the--the--issue is--is--is when can we get--when can we find--find help, for all those who--who--who are waiting?" (I made that up, but you can hear it any day, usually with a lot of "uh" thrown in.) "Is" in particular, being such a common, short, and seemingly colorless word, gets a lot of repetition. So, in sum, people hear "is" repeated in speech, and their minds connect it to wh- type constructions, and they begin forming deliberately constructed sentences with the phrase, "The thing is, is...," which in their minds have thus become grammatical. Interesting, no?
2) "I was reading where..."
Usage from Google:
a. "I had just sat down and was reading where Cary Middlecoff won the US Open..."
b. "I was reading where Alexander Yakovlev's father Dmitri was also involved in the 'mess'..."
c. "I was reading where he had sold about $337 million of Qwest stock..."
Doesn't look like any other linguists have cared enough about this construction to write about it yet (or not that I've found, anyway), and indeed it does strike me as mostly colloquial and not hugely widespread. Some of us, including me, will say informally, "I was reading how..." for any of the above style of sentences. "Was reading where" gets 2,610 hits on Google, which includes many that don't fit this pattern--e.g., "I was reading Where the Red Fern Grows." "Was reading how" gets 912 hits, and again with some misleading ones. For formal writing, we would probably all say, "was reading that..."--or rephrase the sentence altogether.
Still, I think there is a parallel construction for "where" that might lead people to use it in the above manner. If you were in the middle of A Christmas Carol, you might say, "I was reading where Marley's ghost visits Scrooge," and that would be somewhat different from the sentences above, since you would be abridging in a way: "I was reading the part where Marley's ghost visits Scrooge" is the full thought. In the above sentences you can't really insert a phrase like that, and make it work, but all of these sentences strike the ear in a similar fashion, which is often enough to spin off a new linguistic construction.
Linguist disclaimer: By pointing these out I'm not saying that people who use them are Stupid or Wrong. Don't feel you need to adjust your speech patterns or apologize for them. Given the way language keeps changing, these phrases could well become part of the "standard English" of the next generation. I am also not advocating the adoption of these phrases; to be honest, I would instinctively edit them out if I were correcting papers. My aim here is merely that of the linguist: to make note of relatively new constructions and try to figure out where they came from.