New York, New York -- Critically panned 1974 Scorsese flick starring Robert DeNiro and Liza Minelli. Usually looked at as a musical, though I don't personally think it fits the bill. More of a drama that revolves around two musicians and, therefore, includes music. My very favorite film ever.
Down By Law -- Hilarious 1986 Jim Jarmusch film starring Tom Waits, John Lurie and Roberto Benigni. Beautiful black & white photography, typical Jarmusch pacing (read: slow), great script. My second favorite film ever.
eXistenZ -- 1999 David Cronenberg weirdness starring Jude Law and Jennifer Jason Leigh. I know people who hate this one with a passion, but I thought it was very underrated.
High Art -- 1998 drama starring Ally Sheedy and Rhada Mitchell. Usually gets categorized as just being a lesbian flick, but stands alone as a very compelling drama. I've had to practically force people to watch this one, but they've all enjoyed it.
Suicide Kings -- 1997 comedy/drama/thriller/? ensemble piece. Biggest names are Christopher Walken and Denis Leary. Good performances, the thriller aspect is fairly engrossing, and a few genuine laughs.
The Adventures of Baron Munchausen -- 1988, directed by Terry Gilliam. Don't know if it really counts as obscure, but I watched it last week for the first time in years, and it reminded me how much I love it. Just plain fun.
That's all I can think of off the top of my head. Check into Jim Jarmusch and John Waters -- two of my favorite directors, and overflowing with obscurity.