Neon tetras need a 20 gallon as well to fit their schooling behaviour.
Fish suitable for a 10 gallon:
Corydoras pygmaeus, hastatus, or habrosus (10 minimum)
Microrasboras (chili rasboras, boraras, kubotai rasboras, etc) (8-10 of one species)
Ember tetras (consider these instead of the neons) (8-10 of one species)
Sparkling gourami (3-5)
Rosy loaches (6-8)
A single betta
Endler guppies (up to 4 males, females do better in a 20 due to having babies)
Pseudomugil rainbow fish (6-8)
Scarlet badis *often difficult to feed
Pea puffers*needs special diet care
Ricefish
Some killifish
Wasp catfish (includes Asian stone catfish) (6)
Samurai gourami*** exceedingly delicate, expensive, need groups
Chocolate gourami ** very delicate, pricy, needs groups
Bumblebee goby (there are 2 species common, one is brackish water. Can be fussy with food)