I'm no Shmup aficionado, but casually and thoroughly enjoyed the ones I'm about to list and to the best of my memory/ability here we go.
1) Einhander- Square in its prime, while everyone was all google eyed over Final Fantasy VII there was a diamond in the rough and till this day it still sorta is. Graphically mesmerizing and very well paced, great level design with a modest difficulty. If there is any game similar to this please recommend or Square please re-release!!
2) Raiden Fighter Aces- Compilation of 3 Raiden Games with fun achievements, tight ass controls and mayhem without being bullet hell.
3) Deathsmiles- My first introduction to Cave and its bullet hell, amazing it was even was released in the west but so glad I gave it a chance and pre ordered too! Needless to say this game put many modern gamers on to the shmup scene and although the its bullet hell I've never navigated such beauty in my life. In fact my whole family was able to enjoy this game with its great adjustable difficulty!
4) Tiger Heli- My first foray with a shooter and still a classic if not very nostalgic for me!
5) Guwange- Another Cave Shmup that we owe Deathsmiles to for making its way to the west, phenomenal setting and tiny powerful ninja people, fun as heck thanks to the bullet maze and tiny hit detection equals some intense sessions.
6) Cyber Core- if you like obscure insect shmups and TG-16 this game is really fun and not over kill, good for beginners I would assume.
7) Raiden IV- Similar to my comments above on Fighters Aces, except updated visually and luckily Shmups were popular enough to warrant a release here in the west. Easily can rank higher, primo stuff!
(8) Sine Mora- No list is complete without a Suda 51 game for me, loving and supportive of all his work because he simply can give two shits about what people want or think. He is a gem of an artist and video games are his outlet. Sine More is different from what you would expect from Suda but still severely overlooked and under appreciated game nonetheless.
9) Dead Moon- Another TG-16 game makes my list and thanks to a member here I would have never know about this under appreciated gem. Fast paced for a horizontal shooter with arcade difficulty never got too far in it but glad to experience it however, see for yourselves!
10) 1942- Dead Last on my list, but still nostalgia on my part this was in rotation back in the days on my NES along with Tiger Heli, so yea.
Again I'm no aficionado but hopefully I managed a good list for you guys.
If you can see the future while remembering the past, you may just have control of the present.