Nothin' like naming all sorts of shmups..here I go.
Zaxxon - Sega-induced 80's magic.
Viewpoint - Another isometric wonder with insane visuals.
Mars Matrix - I still have yet to see more onscreen bullets than those that fill the screen in this title.
Cannon Spike/Gun Spike - Shmup or Action game, this one was
pure. You'd think that with between two great developers (Capcom and Psikyo) they'd be able to collectively come up with better boss names ("Bio Gorilla Bloody"? "Crazy Machine Jet Tiger"?), but I find them to add to the game in some strange, hardcore way.
Silpheed - An obvious (though I believe unmentioned) choice.
Einhander - A glorious looking and passionately playing shooter from Square, of all people.
Ray Crisis - Another visually appealing, tour de force of light beams and squint-causing flashes.
Sky Kid - Wait!!, it scrolls...right to left? It does indeed. It also has a Pac Man billboard in I believe the fourth stage. Red Baron is too frequently seen these days.
Phalanx - There's actually a cheat code for the Playstation launch title Zero Divide that opens up an alternate Phalanx game called "Tiny Phalanx". Be sure to look up the original box art for some laughs...
Legendary Wings - Say what you will, this horizontal
and vertical shmup was awesome. Certainly quite different at the time.
Armed Police Batrider - An impressive, if forgotten, little gem from Japan.
MagMax - Customizable shooter with a design similar to Legendary Wings.
1943: The Battle for Midway - Huge improvement on it's predecessor, and an excellent early showpiece for the future of Capcom action titles.
Strikers 1945 - By Psikyo, this and it's sequel were both the pinnacle of standard shooter design.
Nam 1975 - Love the numbered ones. The best reason outside of Magician Lord to pick up an AES on launch day.
Expendable - A unique and underrated 3D-rendered Dreamcast gem similar to Cannon Spike.
Mobile Light Force 2 and Castle of Shikigami 2 - Awesome look and hardcore scoring system that takes years to learn (for me it did).
Raiden Fighters Jet - The best of a great series.
Dimahoo - Absolutely insane. A somewhat twisted game visually, I've heard it called the Silent Hill of this genre. Maybe some of the art and a few scenes later on, but the core game is just a richly colored and beautifully crafted experience.
Espgaluda II- Some of the best hand drawn spites in the world, extremely striking. The original is amazing too.
Progear No Arashi- Has anyone else played this? This one was yet another Capcom published Cave gem. It had a neat "newspaper" theme within it, but most notably, it looked and felt like an adventure game as it scrolled along, being that the art was so unique in each area and drawn as though it had to be specially designed to suit the gameplay, which wasn't the case.
Ibara- How could I have forgotten Ibara

Notoriously difficult, requisite shooter.
Shienryu Double Shooting - A cool twist on the game released for the Playstation in the U.S. as Gekioh Shooter King.
Perfect Cherry Blossoms - I've yet to play this one, but it looks inspired and has gotten a good fan base in Japan.
Oh boy...look at the time. I'll lastly mention that full-scroll shooter segment at the end of Gustar Heroes, Trizeal, Border Down, Giga Wing 1+2, Android Assault, Fantasy Zone, Panorama Cotton, Magical Chase/Cotton Original, Phelios (pronounced Fill-ah-sh), Chaos Field (arguable), Metal Black (inarguable), and my least favorite but still essential Yar's Revenge.
I love shmups.
EDIT: After seeing full post, I can confirm the above statement. Oh, Radilgy.
EDIT2: For more awesome shmups, look up the rest of the Cave and Seibu Kaihatsu games; they rarely do a bad one. Nanostray and Iridion 1+2 are good handheld ones, though a Nomad or PC Engine LT will turn some good console shmups into portables as well. Wonderswan does some good too, and maybe the Treamcast if you don't mind it's faults. And of course, the classic essentials/variants of shmup-age: Centipede/Millipede, Space Invaders, Asteroids, Phoenix and Defender. Now I'm finally satisfied.