I heard someone once say that there wasn't much to the SHMUP genre. That's it's just shooting and dodging. I totally disagree. True, a lot of SHMUPs consist of the aforementioned tactics, but there's a lot more to it than that. This is probably the one genre that pits you against wave after wave of enemies, testing your eyes and your reflexes with different AI or power-up systems. Be it the unique AI of Zanac or the strategic gameplay of R-Type, this is a genre I just can't get enough of. I love shooting down enemies and manuevering through waves of bullets. This is a genre where I feel that it really comes down to my skills as a gamer and not just the weapons I've got.
As for my fav SHMUPs, I'm really fond of these:
Gun-Nac
Ikaruga
Gradius III
Gradius V
R-Type Final
G. Darius
Space Megaforce
Thunder Force III
There are probably others but that's all I can think of right now.