(From the many Godzilla related things over the years.)

Godzilla (Showa Era.):

Godzilla (Heisei) (Reserved.): claimed 

Godzilla (GMK.):

Godzilla (2014): me

Godzilla (Monster Planet): Me

Cyber-Zilla: 

Zilla Jr: claimed

Biollante:claimed

Kiryu: claimed

Mothra

Battra: Me

Komodithrax:Me

Gorosaurus:

King Ghidorah

Mecha King Ghidorah

Angirius

Baragon

Varan

Rodan

Sanda

Jet Jaguar

MOGUERA

Male Muto

Femuto

Super Mechagodzilla

Mechagodzilla: 

Black Mechagodzilla

Monster Planet MechaG: claimed

Mechagodzilla II:

Destroyah

Megalon

Gigan

King Caesar

Ebriah

Megagirus

(I'll add the rest.)

Pacific Rim:

Dr. Geizler (I think?):

Otachi: taken

Leatherback

Knifehead

Axe head:

Mutavore:

Slattern:

Scunner:

Raiju:

CKC(not open until Xeno responds.)

Gamera.

Gamera (From the Guardian of the Universe Trilogy.):

Gamera: claimed

Irys:

Legion:

Jiger:

Viras:

Gyaos:

Barugon:

Guiron:

Zigra:

Other: 

Colossal:

Rhedosaurus: