There are literally hundreds of side quests. Some are from random NPCs around the city, and others are part of a larger, side story arc.
The Thief Quests start by talking to the Thief who can be found in the Port by the clean freak's house.
Talk to the Thief, choosing the first option until there are 3 options and the middle one says “できるとも.” Choose this option to start the first quest, the objective of which is to go to the Main Street and steal some towels from a geisha. The Geisha will be marked on your minimap as a bullseye, and can also be identified visually by the 3 Patrolmen following her. When she isn't looking at you (a pop-up in the bottom right will say “今だ！”) run at her straight on. Bumping into her properly will cause her to drop her towel and you can pick it up to complete the quest. If you are too slow, the bottom right pop up will say 遅かった･･･. If you hit from the side, a different message will pop up. Try again until you get it.
Talk to the thief again for the next quest. Just mash the first option until you get it. This quest is to go to the Highway and steal something from the Sumo wrestlers. Quick travel to the Highway and as you are running to the opposite end, pick up all the trash (green bags がらくた) you find on the ground. They give you bonus gold after you complete the mission. Find the treasure (marked on the minimap) and steal it. The Sumo men will chase you, and you can either fight them (20 or so) or run. If you can create some distance by juking them, you can run to the graveyard. Receive 2000 gold for quest completion plus any bonus from picking up other trash.
Talk to the thief again to get the quest. Fast travel to the Graveyard. The mission is to kick the corpse (Hold R1 and press square) until it drops the treasure. Unfortunately, you have to apologize to Buddha after each kick, and there are 3+ guards watching you. One is stationary, staring at the corpse, and the others are patrolling. Bump into the stationary guy to shove him far away. If you can, get him stuck on something so he can't walk back. Then, when the other guards aren't watching, kick the corpse. Eventually it will drop the treasure. Pick it up and run away. Receive 2500 gold for completion.