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War:

This puts you in a war themed setting usually the historical World War (mostly the second one) but can also be a general war shmup. The machines are usually aeroplanes but can also use powerful fighters though the usual norm is going for realisim by using militaristic enemies and fighters.
Capcom's 1942 series
Psikio's Strikers series
Video System's Aero Fighters/Sonic Wing's series
Fire Shark/Same! Same! Same!
Silk Worm
Dyger
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Foreign Planet:

You are put in a foreign planet where you usally fight aliens. Though some mimic earth, other planets look as if they are a giant alien, as if the entire world were alive.
Some foreign planet shmups include:
Exed Exes
Alpha Mission
X Mission
Turtle ship
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Space Explorer/Final Frontier:

The most primitive of the shmups. The usual space exploration and space missions where the main fighting area is in space instead of it being on the way i.e. unlike Hot Dog storm where you eventually get to space, Gradius will always be in space. This may be debetable and could probably fit in the foreign planet categorie but I'm thinking more on being in space, where there is little to no obstacles standing on our way besides the enemy, where as in the foreign planet, you sometimes have to manuver through debris and enviroment.
Some space exploration shmups include:
Gradius
Darius (though some games start on a planet but eventually go to space)
Phoenix
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Earth invasion:

Just like the war themed shmup, you are set on earth but usually with a futuristic ship at a futuristic time and fighting aliens or a hostile organization.
Some earth invasion shmups include:
Asuka & Asuka
E.D.F
Acrobat Mission
Air Buster
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Cute Em' Ups:

Cumps are shmups that focus on bright, cute, cartoonish graphics. They usually involve ridiculous characters but can sometimes have a serious storyline.
Some cute em' ups shmups include:
Konami's Paridous series
Fantasy Zone
Battle Chopper
Konami's Twinbee series
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Steampunk:

The enviroment and overall feel have a non-computer techonlogical feel to it and mainly depend on natural sources to power the machines, mostly steam from which it is named for but not always. The machinery look more mechanical with the graphics having a cold steel feeling to it. Unlike other bosses where they shoot magnificent projectiles, steampunk bosses shoot realistic ammunition all the time.
Some steampunk shmups include:
Steel Empire
Battle Garegga
P-47
Gun Frontier
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Fantasy:

Somewhat different from cumps but the setting is more based on fantasy and fairy tale. The graphics can be bright but are sometimes dimmed yet colorful enough to seperate it from it being too cold. The enemies and players are sometimes based on fantasy settings such as fairies and witches. Unlike cumps, it is not always ridiculous with its enviroment and treats it as if it were a story book.
Some fantasy shmups include:
Chariot
Cotton series
Mystic Riders
Dimahoo
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Mythology:

Just like Fantasy only it is based on mythological themes with enemies, allies, power ups, and projectiles based on mythological figures.
Some mythology shmups include:
Guwange
Phelios
Legendary Wings
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Dystopia:

This category could be debetable. This one involves settings of destruction and chaos but does not neccesarly mean is a setting in itself. Usually the game looks very dark, grim, and sometimes hopeless with everything from the players and they enemies looking dirty and grim.
Some shmups I find that fit this description include:
Metal Black
Last Resort
Nostradamus
Robotron: 2084
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This is the list I have for now but I want to see if it is good or what improvement it needs.
