For me it's a 50/50 toss between Enterprise-D and DS9/Defiant. I think I would serve on DS9. Much as I love the Enterprise under Picard that station is just so interesting. Never a dull moment at the mouth of the wormhole.
The NX-01 for me. Being the first ones out there, doing what no-one else on Earth has done yet, making history (that no-one remembers ). Not all the creature comforts from later ships, but enough there to keep me comfortable. And providing I don't get blown out of a hole in the side of the ship near Azati Prime, I can say I helped save Earth from destruction (which still no-one would remember ).
Most likely NX-01. It would be scary, dangerous but I would get out there to explore the wide expanse of the galaxy. I wouldn't go for a bridge post, I'll probably get chewed out most unpleasantly in front of all the senior officers if I make silly mistakes. I'll apply for a posting in the armory department. At least they are the first ones to run for the defence stations to arm and prepare to repel any attacks. Yes, I know, the ones most likely to die too. But then, that's preferable to painting oneself as a nice fat target for invaders with a flashlight in the dark.
1701-nil under Kirk, provided I won't have to wear a red shirt. More freedom there I think. Second choice is the D.
Seeing as I'm from this century, and I presume that I'd have my family with me, I'd first like for us to spend a few years vacationing as complete wastes of space , taking in the sights on Earth, Vulcan, etc. During this time, we'd also be taking advantage of 24th century medical technology to get healthy and fit. If or when we got bored with that, we'd either set up some sort of business in the DS9 Promenade and meet Gamma Quadrant aliens that way, or maybe even get hold of our own ship and go exploring/trading in that direction. I voted DS9/Defiant, since that's the closest option. But just because we're transported to the future doesn't make me more likely to run off from my family. And families on Starfleet ships is just a dumb idea, IMO. If I were flung forward alone, I'd try continuously to get back. Preferably, with a copy of the 2050 sports almanac.
Deep Space Nine. I don't know that I have what it takes for Starfleet, but there's lots of good stuff for a civilian to do there, and with my retail experience, I'm sure I could come up with something. Maybe I'd see if Garak wanted an assistant. I could learn a new skill, work with a creative mind...and best of all...work with someone who won't crank the air conditioning so far that it makes me shiver and snot all over the place!!!
My first choice is Kirk's ship, as long as I'm not wearing a red shirt. Second choice would be Archer's ship.
The miniskirts on Kirk's watch are pretty spectacular, but I think I stand the least chance of being blown out of the sky on Picard's ship.
Post one for Voyager, I love the Intrepid Class StarShip. In some ways it would suck to be stuck in the DQ, however, there would be new and exciting and different new species and astronomical phenomena that you may never see in the AQ, BQ or GQ...