I always thought it was. I thought it was the equivalent of whose for something that was inanimate. (And I’m not talking about that’s, which is a contraction of that is.)
For example: It’s dangerous to drive a car thats headlights aren’t working.
But apparently not. It’s supposed to be: It’s dangerous to drive a car whose headlights aren’t working.
I’ve been checking grammar guides and they seem to agree that the latter sentence is correct. But it seems wrong to me.
Is it just me?
Yup, just you. It’s not a word. While “whose” sounds like it should be person related, it’s perfectly fine to use in association with a thing. “Who” on the other hand only refers to people. You’re probably riffing off of “its”, which is the possessive of “it”.
Well its colloquial either way, so acceptable in some informal speech, but “that’s” looks wrong without the apostrophe. I do realize that that violates the “apostrophe = contraction, one word = possessive” rule of who’s vs. whose etc.
I’ve never thought thats was a word. But I avoid whose with inanimate objects; I’d usually rephrase “whose headlights aren’t” as “with headlights not.” (I think I’ve often floundered for a whichs that corresponds to your thats 
but as my mouth starts to form “which” I quickly switch it to “with.” 
)
Probably best to just rework the phrasing.
It’s dangerous to drive a car with broken headlights.
No, it’s more likely based on the use of who and that, depending on whether you’re referring to an animate or inanimate object. I guess I just feel there should be two equivalent words for the possessives.
A person who has red hair: correct.
A person whose hair is red: correct.
A house that has a green roof: correct.
A house thats roof is green: incorrect?
Okay, it may not be grammatically correct but it seems to me like it should be.
I’ve never liked this line of reasoning. I figure if you don’t know how to phrase a sentence correctly, it’s a sign you need to learn the rules not to avoid the sentence.
That’s is a word. That + Is. Thats is not a word.
Speaking as a college-educated bloke who still mucks up “its” and “it’s” more than half the time, I can sympathize with the OP!
Say, what about the plural of “that?” “When Joe says ‘that’ he means one thing, but when Will says ‘that’ he means something else. The two ‘thats’ aren’t the same.”
Kind of like "gots’. When I was a kid, I used to struggle with that word. (It drove my Mom crazy.) I had a bad habit of using “gots” when I should have used “has”.
I have no idea how I fell into that bad habit.
Shakes: Heh! Were you in Yosemite Valley twenty five years ago?
My sister and I were there, and, nearby us, were two kids who had just found a Swiss Army Knife someone had lost in the woods. They were exultant. “It gots a knife… It gots a corkscrew… It gots a scissors…”
Ever since then, “It gots” is our family in-joke for “fully-featured.” Like, someone just bought a new car. “Does it gots?” “Oh, yeah, it gots!”
Wow, there must be over 15 ***thats ***in this thread.
Nope! tweren’t me. I think I outgrew that word 30 years ago. 
nm…
“That’s all folks!” – Porky Pig. As authoritative as William Safire and William F. Buckley, Jr. combined.
See, and I figure, if I’ve a perfectly cromulent way of forming a sentence and a rule says it’s wrong, the rule is stupid and should be ignored.
Please don’t embiggen yourself.
It seems to me that “that” is possessive there, so if you have to use it, it should follow the regular rule for possessives and take an apostrophe s: “that’s.” “Whose” and “its” (as a possessive) don’t have apostrophes, but they are irregular. Everything else, AFAIK, must have one.
However, to use “whose” in that sentence would certainly be more standard, and, to my ear, more correct. Although the car that is being references is not a human being, in this particular case it is being spoken of a bit as it were human, and so capable of owning something.
If you must avoid the appearance of even slight anthropomorphism, you can use the semantically equivalent “if its”: “It’s dangerous to drive a car if its headlights aren’t working.”
And, no, that’s not rewording the sentence. “If its” means the same thing as “whose” in that context. The “if” is the conjunctive part, and the “its” is the possessive part. The only difference is exactly what you wanted to avoid–the implication that the car is a person.
Hence inventing a new word is unnecessary, not that it stopped the usage of “cromulent” for “legitimate.”
Consider my other example. You can’t switch “Drive by the house thats roof is green” to “Drive by the house if its roof is green”.