Well?
The second one. There’s more than one “gentleman” and more than one engine.
I think the latter, since ‘men’ is plural.
As opposed to ‘Everyone should take his seat.’
And, in posting from the center of the racing universe (so to speak), lately it’s been:
“Gentlemen and lady, start your engines.”
The first one, of course.
At the time of this posting, “gentlemen start your engine” gets 1,860 Google hits, and “gentlemen start your engines” gets 41,700 Google hits.
The strong preference for the latter suggests that it is clearly the standard version.
That doesn’t make it the correct version.
“It doesn’t mean a thing,” yields a mere 9,960 results, while “It don’t mean a thing” returns 96,400. That doesn’t mean a thing when it comes to the point of which phrase is correct.
In the case of “Gentlemen, start your engines,” the correct form is the most common – but it’s not correct because it’s common. It’s correct because it conforms to the rules of English grammar, such as they are.
I would argue that you are focusing here on a fixed colloquial expression, if not an extremely popular song lyric, in which nonstandard verb agreement achieves exactly the intended effect and is thus correct in context. For the (probably few) people on the Internet using it in earnest, it is correct for them in their nonstandard variety of English.
I wasn’t trying to claim any causality. It’s just that frequency is a reasonable index of acceptability, for obvious reasons. When we’re talking about a choice between two variants, you can appeal to all the philosophy and logic you want, but in the end, only empirical data, such as google counts, serves as evidence. And if you think there’s any practical difference between “grammatical correctness” and “acceptability”, you are fooling yourself.
And how, pray tell, do we discover the rules of English grammar if we reject data that we consider to be “incorrect”?
I figured the second one was correct, as that’s what I’ve always heard. However, lately I could swear I’ve been hearing the former more often. Not being a grammariantologist, I wondered if the singular “engine” was correct because each gentleman, or lady, has only one engine to start.
Anyway - thanks all!
I thought the starter said “Drivers, start your engines.”
My humble O would be
Gentlemen start your engine, if each gentleman had a single engine, while Gentlemen start your engines would apply if at least 1 gentlelmen had more then one engine to start.
narrows eyes You… descriptivist. 
We’ll have to agree to disagree on that point. To me, that’s just an appeal to popularity.
The question is about grammar, and grammar has useful rules to ensure the clarity of communication. In my view, “Gentlemen, start your engine,” is incorrect because it implies that the speaker is addressing a group of people who have one engine between them. It’s incorrect for the same reason that “The group of men assembled at the bar removed their hat in memory of their fallen friend” is incorrect.
To some degree, I agree with you. If we were arguing about whether “To boldly go where no man has gone before” was less correct than “To go boldly where no man has gone before,” then I would be inclined to concede altogether.
Some grammatical rules are more than pro forma conventions, though – they’re necessary for clarity of communication. Extreme descriptivism tends to make a language less useful for conveying actual information. A huge collection of ungrammatical idiomatic expressions isn’t as useful as carefully adhered-to grammar.
If we say, “Oh, it doesn’t matter if people bother to inflect their nouns for number,” then the usefulness of English as a language is reduced. If we have to choose between two phrases, and one is properly inflected for the situation it describes, and the other one isn’t, we should select the one that accurately conveys the intended information.
Otherwise, you’ve got anarchy. Ick.
The problem there is the disagreement in grammatical number, and the confusion about possession. Are you asking for one engine to be started? “Gentlemen, you may each start your engine,” is a fix for that, and is more precise in that it removes any possible doubt about how many engines each gentleman possesses.
I have never, ever heard “Gentlemen, start your engine!”. I hope it hasn’t caught on anywhere.
“Gentlemen, start your engine!” has the sense of “Gentlemen, start your collective engine!”. Many gentlemen, one engine.
On the other hand, “Gentlemen, start your engines!” sounds like “Gentlemen, start your respective engines!” Many gentlemen, many engines. The “respective” is understood, and pared out of the sentence for the sake of catchiness.
And no, it isn’t “Gentlemen, start your respective engine!”. When I mention Bill and Mary’s respective children, that sounds fine. When I mention Bill and Mary’s respective child, it should trigger a “Wha? No, that’s not right.” response in the reader.