Their style, packed with genius edits and visual gags in many shots, translated perfectly to the big screen and is almost a star in its own right. Edgar Wright uses the editing process as a storytelling tool to a much greater extent than pretty much anyone around right now. Sure, there are snappy edits in most films these days, but not with this level of creativity.
The cast is pretty much a who's-who of modern British comedy (check out the scene where the characters are mirrored completely when Shaun and the gang run into Yvonne and her own group for the proof). I think just about the only person missing from it is Mark Heap. Peter Serafinowicz is a marvellous addition to the cast too, coming across as a complete arsehole, which was the point, I guess. I'm sure he's delightful in real life (his Twitter activity would certainly suggest so).
What really sets Shaun of the Dead apart from the rest of the Zombie genre is its humanity. The characters are completely believable, even taking into consideration the comedy and the weird situation. Yes, there's laughs, crude language and pop culture references, but there's also some genuine pathos ("Stop pointing that gun at my mum!" That scene is stunning for the way it turns everything on its head and changes the tone of the film from a romp to a real fight for survival) and moments of real concern for the characters. The tropes of the zombie genre are turned delightfully on their heads, one fine example being the explanation for what caused the zombie outbreak in the first place. The various news channels and whatnot are each cut off during a sentence, which forms whole new sentences while Shaun and Ed are channel-hopping. It's a great idea that gets around the tricky prospect of having to just go for the tried and tested old-school reasons for the Zombie apocalypse. It's difficult to say just how much I love this film.
One thing that I do regret is not following up the call to arms that was posted on Ain't It Cool News back when Edgar and the gang were setting the film up. A casting call for zombies was put out, and I didn't go! Bah. Anyway, that aside, Shaun of the Dead takes pride of place in my collection, and I'm sure the collections of many other genre addicts out there. It's a delight from start to finish, and in every way it succeeds. Comedy, horror, real characters, a great soundtrack, brilliant direction, great editing and completely deserving of every bit of adoration that it gets.