I'm not very good at shmups, but they're pretty enjoyable if I'm in the right mood. My favorites are:

Fantasy Zone (SMS)
Einhander (PS1)
Galactic Attack (Saturn)
G Darius (PS1)
Gekioh Shooting King (PS1)
The Guardian Legend (NES)
Ikaruga (Gamecube)
Internal Section (PS1)
Nanotek Warrior (PS1)
Shooter: Starfighter Sanvein (PS1)
Silpheed (Sega CD)
Tempest X3 (PS1)