Even if you're terribly concerned about story, you can skip the Specials as I have; I saw it in:
OVA
Season 1
Season 2
Movie
I do suggest that you actually see the movie before starting Season 2 as the movie does fill in some background information.
The reason I did not see the Specials is because it is no longer licensed so I cannot buy it, and there are no fansubs that I can find to watch it online. Full Throttle is the same problem; only the first two episodes on veoh were subbed. My knowledge of the Japanese language still isn't good enough to be able to follow an anime without subtitles.
Anyway, I think one of the all time greatest moments was the part in Season 1, episode 13, "Big Trouble at Little Onsen" when the two guys intruded in Aoi's room. I think, bar none, that was the all time funniest thing I've ever seen in an anime. Coming in a close second was the demasking in "Santa Clause Panic". I just felt that toward the middle of the series, the writers seemed to be running out of ideas so they started reusing some of the jokes, like the ones I just mentioned. Still, on the whole, You're Under Arrest! season 1 is still one of the all-time best animes I've seen and it's probably the best comedy I've seen.
Season 2, I felt, was well-animated and I like how if offered more character development, but I felt that some of the episode ideas seemed recycled from the first season. Episode 17, "Curse on a Midsummer's Night" reminded me of "Bokuto Station Ghost Story" from the first season. The last episode, "A Hot Spring, a Yukata, a Confession" reminded me of "Big Trouble at Little Onsen".
A final bit of YUA! advice, skip No Mercy!. As much of a fan of You're Under Arrest! as I am, I think No Mercy! easily ranks among the worst anime I've ever seen. It's a side story, so you don't need to see it to understand the main YUA! lore. You can check out my
review so you know exactly what I mean.