Insects and Other

Sodatete! Mushiking!
そだてて!ムシキング!

One of the most well documented oddpets on this site, the Mushiking is a very detailed and very fun little pet. Raise your beetle, feed it a variety of items, equip it with stat boosters, and fight in a tournament!

The shape of the shell determines what type of beetle you can get. Stag Beetles (family Lucanidae) come from the most common pincer-style shell, Rhinocerus beetles (subfamily Dynastinae) come from the shorter horned style shell shown under the Stag shell on the left, Atlas beetles (genus Chalcosoma) come from the three-pronged shell.

The two-pronged shell where the pincers are shaped differently have the same roster as the Stag shell, with the addition of the Hirata Stag beetle.
The translucent blue shell with the yellow X is a special anime shell, but has the same roster as the normal Rhino shell.

Detailed fan-made instructions here.

Ants Life Studio

This is the joy of having an ant farm without having any actual ants!

It's fairly hard to maintain and needs frequent attention to help ward off predators.

Punirunes
ぷにるんず

A fairly new pet that you can actually pet!

Despite being so new, there's not a lot of information in English about this pet.

Cyber Baby

Actually not a baby. Honestly... I'm not sure what it is! It has multiple growth stages. The baby stage lasts over a day (and it hatches sick, for some reason) and dies fairly easily... to an extent. I haven't quite figured out yet and so far have two failed runs. Oops!

Love Chu

An interesting little critter with a connection function. The Love Chu 2 seems to have a bit more refined gameplay than the first.

Hana Hana Mi

A plant!
That's all I know.

Heartune

It seems to be a little angel heart pet.

Pocket Pal

Dinkie-style gameplay, and Tamenagerie says it changes form each day.

Micalun & Devitan

Not entirely a pet, but you do stop the two from fighting and play games.

Unknown

Tamenagerie has the only one of these I've found, and there is no other info on it. Seems to be a car pet!

My Sapling

Reacts to light, sound, and movement, and grows faster the more you play with it. Very unique!

My Pet Messenger

Released by Scholastic.