I’m 25, for the record. I feel like it’s been much more commonly used in the last year or 6 months, to a degree that saddens me. I’ve heard Condoleeeeeza say it in several interviews, all of which open the door to images of her as a donkey following a carrot. She’s sort of equine, isn’t she?
That was unexpected. And awesome.
I’ve never heard the phrase before.
Cheesesteak didn’t have the guts! He was afraid that he would get hit. With a stick! And he didn’t have the nerves to ask for a carrot either.
Well I demand both. Especially the carrot.
I fucking hate carrots. You could put a carrot on a fucking stick and throw them both off the edge of the Disc for all the fuck I care.
I don’t care if they’re raw, cooked, or minced up with raisins in that insipid salad that somebody has to bring to every fucking church picnic I’ve ever been to.
Carrots suck. They have the texture of the rope I had to climb in Mr. Nilles’ gym class when he wouldn’t even put mats on the gym floor, they have less flavor than old saliva and they’re ugly.
Fuck all orange food for reminding me of carrots.
That’s no kind of reward to me at all. You want to reward me? Promise me I’ll never ever see another fucking carrot in my food ever again, that’s all the reward I ask.
Mr. Bus Guy: I love you. I can only deal with a stewed carrot. You know Mel Blanc hated them too? He used to have to chew one when he was doing Bugs and always spit it out after the take. Couldn’t swallow 'em, couldn’t stand 'em. He tried chewing other veggies but nothing made a sound that was “carroty” enough.
“Carrot and stick”, with “carrot” as the reward and “stick” as the punishment, has been around for going on 60 years now:
No! Don’t take away my “herding cats”! I’ll be left with only “nailing Jello to a tree”.
Besides, I love that EDS commercial. (http://video.google.com/videoplay?docid=-8215789355288508566&q=herding+cats)
Well I understood it to be a unattainable reward for a mule. I think a carrot would make a tasty treat for such a beast.
I think this is the phrase “to dangle a carrot” in front of someone, and doesn’t involve a threat. At least that’s what I’ve always heard, in the same context that you have.
And vetbridge even moreso. 
I liked John Carpenter’s version of The Thing better than than the walking carrot B&W flick of the 50s. I still like the carrot from the 50s, it’s just that Carpenter did it better, ya know?
And it even had stick legs. The Carpenter Thing. As a head. A disembodied head. With stick legs.
Okay I’ve just Googled “Condoleeza Rice” and “Goat Rodeo.” Nothing. Would you care to explain yourself?
And what praytell do you dangel the carrot with, well besides a string (and we all know that a carrot and string has sexual conatations in certain circles, so we won’t go there, would be a stick.
Carrot sticks - crunchy!
I haven’t noticed a recent increase in the use of the very old phrase. I’ll try not to use it here (SDMB) so that I don’t needlessly annoy you.
Interesting, I’ve always known it as “carrot on a stick” and it has conjured up images of a rabbit wearing a hat. Extending from the front of the hat is a stick, and dangling from the stick is carrot. The rabbit can always see the carrot, he can smell it, and his little rabbit paws can just brush it if he reaches out as far as he can. But he can never get the carrot, poor wittle wabbit.
YES! I have mentioned this before only to look up at a wall of perplexed faces, or to receive response to the effect “but what’s the punishment?”
The odd thing is that there seems to be considerable etymological scorn poured upon the phrase ‘carrot on a stick’ - it seems that for some people, it won’t be enough to declare that it arose later than, or as a derivative of, ‘carrot and stick’ (either of which may or may not be true) - no, they want the phrase ‘carrot on a stick’ simply not to exist.
-And not in a manner of simple annoyance or overfamiliarity as expressed by the OP; this seems to be a case of them declaring that there is no such idiom - what you just said doesn’t exist.
Well, yes, it fucking does exist, because I just said it, because I’ve been saying it all my life, because I’m not alone in saying it and becauseI’m going to keep on saying it.
I had thought, before this thread, that the threat-free version from my childhood was “correct”, and that the “carrot and stick” variant was a modern bastardization based on misunderstanding. The cites provided by others above now leads me to think that both versions have coexisted for a long time. I can clearly remember the picture of the boy and the donkey with the carrot hanging from the stick.
Me too; I’m not even particularly bothered whether or not it predates ‘carrot and stick’ - fact is, it exists; it’s out there and I don’t see why it should go away just because a few vociferous people with sticks (and possibly carrots) up their arses can’t get to grips with it.