Tyrian, of course. Do you know that the creator is going to put it on the net as a free download? I simply love Tyrian. It has excellent plot, [for a shooter... and a good one overall] humor, explosions and space fleets. And easter eggs, too. And tourists... oh the tourists.
Metal Slug series is very nice... I, personally, prefer the first three ones, with a special emphasis on the first one. It just seemed a bit less over-the-top and more like a war, along with awesome endings. The fourth was nothing special, and the fifth was horribly incoherent. [and no Morden! HERESY!]
Raptor is a nice thing. Excellent graphics and gameplay, but the story is not even touching Tyrian's. OTOH, Raptor has less cartoony graphics, and is generally gritty and dark.
Raiden 2 is good, the deformable terrain is quite nifty, and the gameplay is fun.
Major Stryker is a very good game, with fun gameplay and "3D" in a way [the game was 2D, but you could be attacked from below, see enemy tanks moving on the ground when you are fighting in clouds etc]. Bosses are interesting, and the game is not something unholy in diffulcity. It also has great inter-level talk, and not in a "All Your Base" way. Go and play, I don't want to spoil the fun.

HOTU should help you with finding it. [*cough*the game is about a decade old and unsupported by the publisher, guess what help it is*cough*]
RayStorm is fun, too, because of... 3rd level, IIRC. A massive space fleet. That kicked SO much ass. The rest of the game is good, too, but not as good. Includes extremely powerful last boss.
Banshee for Amiga... the gameplay, the gore, the graphics... a fine combination. It even has a kickass space level, with tiny alien cosmonauts getting splattered all over the space. Whee! Unfortunately my current Amiga won't play it, and the files I downloaded for emulator are messed up - the sound won't work and the last level [the said space one] won't load.
R-Type... because it's R-Type. What more can be said?
That's pretty much it for "pure" SHMUPs. There are also things that are borderline SHUMPs, like Gun Metal.
So bloody what if some of them are not for consoles? I like them.

Many of them ARE console games, though.