Bug Catching

A pastime for kids (ages 5–10) to do during the summer. It's kind of like fishing to see who catches the most and release at the end of the day. Bug-catching nets are a must.

Its mostly a boys' and Tomboy thing, although there are a few surprises. Extremely popular in Japan.

Anime and Manga

In Haruhi Suzumiya, competitive cicada catching is one of the Endless Eight activities. Though, as Kyon points out, this is a weird activity for kids their age (high school).
Yotsuba&!'s Cicada hunt has appeared in a few chapters; she released them inside the house. Currently the page image.
Ichigo Mashimaro
One Gintama episode centers around beetle hunting for money.

Film

In Lucas starring Corey Haim, the titular boy spends his summers out in the woods, catching bugs, tadpoles, frogs, crawfish, and other creepy crawlies, just for the fun of it.
Looks like one of the only hobbies Shilo from Repo! The Genetic Opera has while imprisoned in her room.

Literature

One of the summer pastimes undertaken by the children in Stephen King's It is catching insects, frogs and other creepy crawlies in the Barrens.
In the Dr. Seuss book Summer, one of the joys of summer for kids is catching fireflies in jars.
In Tom Sawyer, Tom and his friends regularly catch frogs and snakes just for the fun of it.

Newspaper Comics

Calvin and Hobbes often go out looking for bugs or "anything weird."
FoxTrot: Jason and Marcus often do this, so as to have a ready supply of critters with which to annoy Paige.

Video Games

Bug Catchers in Pokémon are explicitly based off this. The creator, Satoshi Tajiri, was even an avid Bug Catcher himself when he was a kid, and this pastime is actually what he based Pokemon off of.
You'll spend a lot of time doing this in the various incarnations of Animal Crossing.
The bug net in some of the Legend Of Zelda A Link To The Past is reminiscent of this (you can even catch bees and other insects with it as well as the fairies that return in later games). Later games even have bug collecting subquests (particularly The Legend of Zelda Twilight Princess, where there was a massive variety of bugs to collect and one of a kind gold versions of each you had to collect as part of a side quest).
- In The Legend of Zelda Skyward Sword, bugs have to be caught and given to an NPC in order to improve Link's potions.
The player does this every once in a while in Monster Hunter as well.
In The Sims 2, Sims can catch bugs. They even get a collection box with pictures of the species they've caught. Sims can also catch bugs and put them on display in The Sims 3.

Western Animation

In SpongeBob SquarePants, SpongeBob and Patrick go jellyfishing all the time.

Real Life

The naturalist Gerard Durell's autobiography, My Family and Other Animals, details the author's childhood (and continuing) obsession with nature, and the many different species he collected as a child.
During the Boer War, one of Robert Baden-Powell's favourite cover stories when he was spying was as a butterfly collector. Not only did this seem so innocent to the Boers seeing this grown man chasing butterflies that they didn't realize that he only seemed to find ones around their fortifications interesting, but he could disguise his maps in the pictures of the ones he caught.