I've seen a number of independant films over the years, most of which I can't recall off the top of my head, but some are quite good.
Bubba Ho-Tep is fun if you're a big fan of Elvis, Bruce Campbell, Aussie Davis, or cowboy mummies terrorizing a nursing home. Also surprisingly down to earth in terms of a commentary on how we treat the elderly in this day and age.
Shaun of the Dead is also a very fun movie, particularly for people familiar with classic zombie movies like Night of the Living Dead, etc. And, like Bubba Ho-Tep, has some surprisingly heartfelt moments in what looked to be a silly comedy. There's some really good stuff in there, and it's just smartly written and very well-acted.
King of Hearts is actually French, I believe, so I'm not sure if it'd count as an independant or a foreign film. Anyway, it's basically about a Scottish soldier during WWI who's sent on a suicide mission to detonate explosives set by the retreating German army in a small French town. The town's been evacuated because of the bombs except for the inhabitants of the local insane asylum who get lose and take over. Wow that sounds really bad, it's not a scary movie, it's a comedy– the crazy people aren't dangerous, they're just detached from reality.
The Pillow Book is a very strange movie but visually stunning. I wouldn't reccomend it to people with problems with nudity (male and female), homosexuality, erotica, and some acts of violence and disturbing mentalities. The plot's a little difficult to explain, but it's basically about a girl, Nagiko, who wants to be come a writer but can't get published. She starts writing on people's skin, as sort of a holdover fetish from her childhood when her father would write on her face for her birthday. She meets Jerome, a British linguist, and he agrees to become the "pages" for her book, and she spends a good part of the movie writing on him. Anyway, it's not a film for everyone, but the cinematography really is gorgeous and I find it interesting. The characters aren't typical cardboard cutouts and they're all pretty ruthless and messed up in their own ways.
Shallow Grave is another movie with strange, rather unpleasant characters, but they're put in such a horrible position that you are scared for them. This movie scared the hell out of me when I first watched it, but it's utterly compelling to watch. The main characters aren't exactly the greatest people in the world, but for some reason you find yourself rooting for them to make it out alive and you're not entirely sure why. Plus, it's just a danged effective thriller in its own right.