(Warning! I use MSIE6 at home, and this page crashed the browser)
I like the author's serious attitude, because there aren't many shmups with well-developed stories. Instead, the developers mainly concentrate and succeed in creating the general atmosphere - we all know, remember and love the gross intestine-fest of X-Multiply and Lifeforce, the creepy horror of Philosoma, the biopunk of Bio-Hazard Battle, the epic feel of Spriggan, etc. But what about the actual stories? Here are three shmups and one run-n-gun off the top of my head:
Tyrian - aside from being the best DOS shmup I've ever played it has very interesting and complex plot remotely reminiscent of Star Trek Insurrection
Gaiares - yeah, Telenet Japan once again brings their magic of creating 90s style anime cutscenes that sometimes even make me shed a nostalgic tear
DonPachi series - why not, actually? Their story is kind of creepy too, because I think the fictional world should be really fucked up to allow the rookie soldiers to slaughter their comrades without remorse in training purposes

Midnight Resistance - this one relies more on OTT somewhat campy stuff. The image of two soldiers fighting off the jet fighter squadron with their machineguns makes this game beg for a movie adaptation
Hopefully someone can name other shooters with well-developed stories beyond 'blast off and destroy the evil [insert name here]?'