Mayhem, as its name suggests, is an activity that requires you to do as much damage as possible in the allotted time. The only catch is that you have to use the tool provided to you, such as a specific weapon or super power. The game differentiates the Mayhems into four categories: On-Foot, Tank, UFO and TK (or Telekinesis).
On-Foot Mayhems are further separated into a few kinds, although the game doesn’t make them known until you actually do them. Some of them will give you a specific weapon, while one or two others will require you to use the Stomp power.
One very easy way to finish the On-Foot Mayhems with Gold is by purchasing the Tornado upgrade for the Super Sprint power. Should you do that, simply run around on the sidewalks to build up a huge combo and a high score.