Now, although I got a factual question for the Teeming Millions (and possibly Cecil) to consider- one with a cut and dry, unarguable answer (fucking gin!), it seems that some people would disagree with me.
So, as the title asks, if I go up to my friendly neighborhood bartender and say, “I would like a martini, please,” what should the glass she hands me be largely comprised of?
Yep, gin, preferably Bombay. However, I always order a Vodka martini.
Depends.
I like Bombay Sapphire with a twist, or Tanqueray with an olive or an onion (which technically makes a Gibson, not a Martini). The Sapphire’s a little lighter and more complex, to my mind, and the Tanqueray stands up better to the salty or sour flavor of the salad.
“Very dry Bombay Sapphire Martini, straight up with a twist.” It’s absolutely lovely.
Anyone who just orders a Martini and expects vodka is a troglodyte. 
A martini is made with gin, so when you order a martini, that’s what you should get. But I’ll bet you a gazillion, uh, olives that the bartender will say – “and what would you like in your martini?” Because otherwise, some of the aforementioned troglodytes will likely rebel when served an actual martini. So the bartender will ask before pouring.
(Which is where I say, Bombay Sapphire martini, dirty, three olives, thanks.)
Gin of course. With a swhack of olives, and a slop of olive juice. I’m a dirty, dirty girl.
Further, if you rinse the glass with a touch of scotch first, it makes them go down smooth.
Finally, they should be neither shaken, nor stirred - the gin should be poured over ice, a TAD of vermouth should be added, and then the act of pouring through the strainer is all the mixing need.
Personally, I’m a Beefeater type of girl, but YMMV.
Troglodyte checking in… 
Make mine ice cold Absolute with a dash of Lemoncello, shaken.
For my date, an Absolute with Khalua and Frambois liquor.
GIN.
(Unless it’s a Texas Martini from Chuy’s, which is really a margarita.)
I’m not so much asking preference among drinks that someone has maybe referred to as a martini at some point, I’m wondering, basically, if “martini”=gin and “vodka martini”=vodka is still the norm.
I’ve taken a little survey of my associates, and the conclusion we’ve all come to is if you don’t specify… you get gin. Otherwise we’d need a special breed of psychic bartenders.
Not that that wouldn’t be handy…
Mental note.
GIN!!!
Because I’ve had this happen to me. I did think this was understood. A martini is a martini…a vodka martini is a vodka martini. Which might be good enough for James Bond, but I wanted a real one. And got something (I hesitate to call it a drink let alone a martini) that was half vodka and half vermouth. Bleh.
The glass should be composed of fused silicon. Anyone tries to hand you a martini in a plastic glass, back away slowly.
Well- classicly- Gin, as has been said. But now it’s going over to vodka.
And a martini used to be a cocktail, with Gin AND Vermouth. Now, the vermouth is basicly a non-issue. Most ‘recipes’ for "extra dry martinis’ are really just a funny way of saying “straight gin”.
Now, most martinis are straight, ice-cold vodka with an olive. The dumbing down of America continues…
Vodka, baby. Any kind will do. It’s the effect, not the brand, after all! Bit of vermouth. At least two olives. And keep 'em coming. 
You had me till you said Beefeater… well, that and the scotch and the not shaking, okay, you had me at dirty 
FordPrefect, neck deep in a damp, dirty Bombay martini, and loving it
I own two plastic cocktail “glasses”, for traveling. Being 1000 miles from home and being able to pour martinis into actual unbroken cocktail glasses is worth the heretical use of plastic. The Gods of Gin do smile on the intention rather than the deed. (or the spelling)
My theory is that this is because the most popular vermouth, Martini & Rossi, sucks.
I used to be an “I love Martinis–the cute glass is a neat way to serve Bombay Sapphire” kinda guy but then I picked up some Noily Pratt (I’d been using their sweet vermouth in Rob Roy’s, another favorite cocktail, for a while) and I must say it’s an enormous improvement. I actually enjoy martinis made with the “classic” proportion of vermoth now.
I gave up Noilly for Martini & Rossi… interesting.
I don’t drink them, insofar as I have trouble distinguishing gin from fingernail polish remover, but yeah, a martini is gin.
Gin drinkers, being understandably embarrassed about the fact that they’re drinking gin, like to pretend they are drinking a mixed drink, called the martini, but you always hear them saying things like “just pick up an empty bottle of vermouth and bring it to the general vicinity of the martini glass and let the vermouth vibes permeate the gin, that’s sufficient” or “pour two tablespoons of vermouth into a martini glass, and then pick up another martini glass and fill that one with gin and stick an olive in it and hand it to me”, so who are they kidding?
::shudders at the thought of drinking gin::
Clearly the answer is gin…and unless you specify, it will be cheap gin.
For me, the question here is whether your glass will have any ice in it. I’ve been told, and it was my unfortunate experience ONCE, that without requesting the drink be served up, it will be served on the rocks. This runs contrary to what I think would be the typical expectation of someone ordering a martini. I would guess that the vast majority of martini drinkers prefer the drink to be served up.
In a way, these questions are moot, as any establishment worth drinking martinis in, will ask to clarify a generic order of “martini”.