The goal of Mayhem is to do a certain amount of property damage before time runs out.
For the duration of the activities, all weapons have unlimited ammo. There are bonuses effect either for using a certain weapon or destroying a certain type of car, as well as for destroying various furniture in the activity-specific area.
This activity has an area limit, but destroying things outside the borders is still counted as long as The Protagonist remains in the radius of the area. Notoriety increases during gameplay, but is removed after exiting the activity. Combos are used to increase the amount of money received from destroying items. There is a timer for combos, which increases when more points are earned.
The timer stops when outside of the area, and that collision rules apply; only items that can be damaged outside of the activity may be damaged inside.
I got a little work for you. Until I opened this club, I was a nobody. My neighbors, my family...they didn't give a fuck about me. But as soon as I had money, that all changed. Suddenly everyone's my friend, and they all need favors. Now, I ain't got a problem helping people out, so long as they understand the consequences. If they meet their end of the deal, there's no problem. But when they don't, that's when I need people like you, Holmes. Got out there and show those pendejos why they can't fuck with Rico Martinez.
Come in. It's a pleasure to meet you, I'm Mary. You may be wondering why a woman my age is running a place like this. I'll spare you the details, but suffice to say, my ex husband didn't want me to have the club, so naturally I took it. Now that we're divorced Nathaniel thinks he can run off with my trollop of a sister. Sadly for my ex, this isn't the case. Nathaniel owes me thousands of dollars in alimony, and I intend to collect. Until he pays me, I want you to go and destroy as many of his holdings as you can...the more you hurt him, the sooner he'll pay.
Sup Sup yo. I appreciate you commin' round on short notice...I got some shit that needs to be worked out. Some cop just transferred to town and he ain't payin' me any respect. The bitch came in here runnin' his mouth tellin' me he was gonna be put out of business. What that Uncle Tom nigga didn't know, was I got a couple cops on the payroll and they told me the raid was tonight. I want you to go out there and stir up so much shit the cops gotta look elsewhere...while they on yo' ass I can hide all the K. I know it sounds risky bro, but don't worry, you'll get hazard pay.
Take a Bulldozer to the Mayhem area, and run over anything that's linked, such as a chainlink fence. Since the Bulldozer is already extremely tough and police can't pull The Protagonist out, at least two levels can be completed before needing to get a new one.
During Mayhem, Anchors often appear, perhaps in attempt to record crimes. They are worth more points than other vehicles.
Super Speed makes this trivially easy, as all you have to do is keep charging everything destructable. Find a busy street and sweep up and down it repeatedly. Streets with parked cars are also very good; you can rely on the parked cars to spawn whether traffic is busy or not.
With the Tornado power you leave a wider swath of destruction, hitting victims you missed a direct hit on.
If you don't have Unlimited Sprint, you may want a weapon with a wide area of effect for when you need to recharge stamina.
If you don't have No Ragdoll or Immovable Object then it's even more important to keep moving to distance yourself from explosions.
Look at the entire map to see where the high value targets are arranged, and choose your course to line them up in order. It's a good idea to build up combo first, to multiply the bonus points for destroying them.
Mayhem was removed from the Japanese version of Saints Row 2, along with Hitman and Human Shield.
Mayhem is not included in the German version of the game.
One easy way to earn combos in Mayhem is to blow up long lines of fences. Saints Row: The Third makes fun of this - Kinzie assigns The Protagonist to blow up a fencing company, and the Achievement/Trophy for completing all Mayhem instances is "Fence Killa 2011".
During Hostage in Saints Row 2, one of the male passengers references Mayhem with the random line "You don't get points for knocking stuff down".