Bart has a stamp collection that was stolen in the episode, Homer the Vigilante.
When Homer is told he is a bad parent we learn that one of Bart's hobbies is building and racing soapbox racers. Homer tried helping Bart build one, but it turned out poor and dangerous. In the end, Bart drove Martin Prince's car, as he had broke his leg in a crash earlier.
Getting into mischief is a more popular hobby of his. This is a pastime that we see him do frequently and he has caused many problems with it.
He likes to play with his slingshot.
Skateboarding is one of his favorites. He is always using his skateboard to show off or try daredevil stunts. This is also seen at the beginning of every Simpson's episode on the intro. One time, he wanted to show off by jumping Springfield Gorge on his skateboard. Every kid in Springfield was there to watch, but Homer wouldn't let him, then he rolled away on it but hit the cliff on the other side.
Bart enjoys doing prank calls to Moe's Tavern. He asks for a persons name by saying the surname first. The names always makes Moe look like an idiot, for example: "Ivonna Tinkle"
He loves watching Itchy and Scratchy which is introduced by his idol, Krusty the Clown. Every episode has him and his sister, Lisa in tears with laughter. They also visited Itchy and Scratchy Land, a large theme park based on the characters.
Collecting comics is a hobby of Bart's. His favorite is Radioactive Man. The comic was to be turned into a film based in Springfield and Bart applied for a part, but Milhouse got it instead.
We frequently see Bart annoying his sister, Lisa. It could be just instinct, but it doesn'thurt to call it an hobby.