The beautiful part about Ikaruga on the Dreamcast (I don't know about the Gamecube version) is that you can set it up to run vertical (rotated) on a TV or montior. This also holds true for Gunbird 2 on the dreamcast, though that's available on MAME as well.
Ikaruga is usually mentioned alongside Radiant Silvergun as the pinnacle of the hardcore Shmups. Radiant Silvergun was available on the Sega Saturn, but I have yet to find a way to play it myself.
Check out shmups.com for more selections and recommendations. Sometimes they reference shareware shmups for the PC.
As a side note: If you want to see more shmups released in the US, buy the shmups that are. Show'em there is a market here. Off the top of my head, recent releases (last year or so) for current consoles:
Mobile light Force - Playstation
Mobile Light Force 2 - PS2
(these are translations and re-names of two decent Japanese shmups, and are a good bargain. I believe you can get both new for under $20 combined)
R-type final - PS2 (no, it's not vertical, but it's a good shooter anyway)
Ikaruga - Gamecube
Gradius 3, 4, 5 - PS2 (again horizontal, not vert)
Any others people can think of?