See, wombat, with your aversion to violence, I was thinking maybe March of the Penguins would be the movie to suggest, but then I remember how I traumatized a couple four year olds because I forgot the part where the almost hatched baby penguins die, and also there's that one shot of a seal grabbing a penguin... But it really is worth seeing, and doing the work you do, you are basically required to see it. Seriously - if I want Morgan Freeman to follow me around and narrate my life.
I also vote for American Splendor and The Royal Tennenbaums/Life Aquatic and Being John Malcovich. None of these are violent or sex heavy.
When it comes to the ones with violence, depravity or sex in em' I have to agree with all of the above, especially Fight Club and Pulp Fiction. I'd also add The Birds, Eraserhead, Blue Velvet, A Clockwork Orange, The Crying Game, The Godfather, Psycho, , and Scarface to the list (sorry, some of these aren't all that recent, but they get referenced like crazy and are great).
Not that referenced, but still great are Edward Scissor Hands, The Professional, Bowling for Columbine. Also, A Christmas Story never makes anyone's must see list, I think because they only think about it at Christmas... but it's just perfect.
Also Magnolia, Amelie, and the Hours are must see, but Tech Support will probably not want to watch these.
As for more recent culturally referenced movies, Broke Back Mountain (although I found it boring and contrived in parts) has been, and will continue to be referenced to death. Same goes for Pirates of the Carribean, which I couldn't stomach at all.