Tyrian 2000 for DOS, please.

http://www.youtube.com/watch?feature=pl ... CFFn336RN4

Probably, hands down, my favorite SHMUP I've played on a home computer.

Tyrian is ace. There's no reason not to play it either as it's free on gog.com

Shmups I keep returning to :

Xevious
Phoenix
Galaga
Super Aleste (snes)
Galactic Attack aka Layer Section aka Raystorm
Parodius series
DoDonPachi
Darius Gaiden
Gradius V
Gunbird 2

.......and I still believe that Asteroids is one of the finest games ever created - no matter what genre.

This was a terrific little PD shooter for the Amiga that appeared on a coverdisk, Transplant:

http://www.youtube.com/watch?v=sXQW7q1fOOk

It eventually evolved into a full price game, Seek and Destroy, which I've never actually played:

http://www.youtube.com/watch?v=_eHUYicOAIw

Looks good, though.

If we are talking PD games then I must mention the sublime ST game Cybernetix. Shame the sound is a bit crap on this video though:

http://www.youtube.com/watch?v=CNvVSYCd4Og

Yup, that also appeared on the Amiga, and you're right, it was a top little shooter. I had the AGA-only full-price incarnation, Overkill:

http://www.youtube.com/watch?v=sKxEuGvzXYQ

In all honesty, it was no better than Cybernetix. It was somewhat purty though.

A couple of great ones from the Spectrum:

http://www.youtube.com/watch?v=CbXu3pseYwU

http://www.youtube.com/watch?v=J6CltV7CRsA

The first game, Light Force, is notable for its full colour graphics without any clash and plays as well as it looks. While Firebird's Star Pilot is all in monochrome but has one incredibly unique feature, a split screen 2-player mode!

Okay.

Slap fight
Gynoug
Hellfire
Dragon spirit
Thunder force IV
Zynaps

Oh yeah, and Deep Blue 

The NEC TurboGrafx has a lot of epic shoot 'em ups, the best one has to be Blazing Lazers (also known as Gunhed in Japan):

http://www.youtube.com/watch?v=OquO7XKHDT4

But one of the more under rated shooters that I quite enjoy for the machine is Psychosis (also known as Paranoia in Japan):

http://www.youtube.com/watch?v=PzrMzO9vFFA