Patrolling the city is a classic element of superhero comics. Not all superheroes patrol, of course - many wait until they get news of a crime to follow up on it, via radio or tv reports, and some superheroes are well-connected enough with the local government that they rely on being alerted to any crimes as they occur.
That said, some heroes prefer to be a little more pro-active in their heroics, Batman being the classic example with his nighttime patrols. It can be a valuable tool for a hero in a sandbox superhero campaign, running across villains, encountering clues that can perhaps lead to something greater, or just running across lesser problems that still need dealing with (and perhaps picking up some extra XP for the hero in the process).
The GM for a superhero sandbox campaign needs to determine how finely detailed they want patrolling to be, and as a result how many dice rolls they're willing to put up with as a result. My preference is for breaking things down into four-hour blocks, but you could probably narrow things down to individual one-hour blocks if you preferred.
Determine what part of town the hero is patrolling during a given time-block, and then see if that particular neighborhood falls under the Area of Operations (AoO) of any villains or organizations to determine any Major Encounters. If no villains or organizations operate in the area, there will be no major encounters, no matter how long the hero patrols there. Otherwise, there is a cumulative 1-in-10 chance for each villain/organization's AoO of such an encounter (for example, if two solo villains, a villain group, and an organization all have AoO's in a particular neighborhood, then there is a 4-in-10 chance of a encounter with one of them during a given patrol). If more than one villain or organization is in the area, determine randomly which is encountered. No matter how many villains or organizations are present, a roll of a 10 on a d10 indicates no major encounter during that patrol. Note that the time of day may affect the odds of such an encounter; if a vampire is operating in the area, for example, their AoO would not modify the encounter roll if the hero insists on patrolling during the daylight hours.
During a major encounter, determine if the villain or organization would prefer to attack the hero, or would rather continue on unnoticed. Most villains will be aggressive, but some would prefer to commit their crimes unnoticed, or might not want to stop to beat up the hero as they're on the way (to their minds) to something more important. Most organizations will prefer to operate without drawing superheroic attention, but if there is bad blood between the hero and the organization, it's possible they may be laying in wait to attack the hero, especially if said hero has made a habit of patrolling this particular area. If the villain or organization is aggressive, assume they are trying to ambush the hero (give the hero a chance to notice what's about to happen, based on how your individual game system handles this sort of thing). Otherwise, the hero has spotted the villain or organization in the middle of some crime (or on the way to or from same), and can either try to stop the crime in progress, or can attempt to follow them back to their base of operations.
If a major encounter does not occur during the patrol, there is also a chance of the hero encountering a Major Clue, but only if they are looking specifically for a villain or organization that operates in the area, or hunting for clues to a crime tied to a villain or organization in the area (even if the hero is as yet unaware of the connection between the crime and the villain/organization). Roll a d10, a modified 10+ indicates a clue of some sort is uncovered, allow a +1 to the roll if the hero has a skill or background relevant to the search (electronics skill while hunting an electrical villain,or a military background while observing an organization recruiting ex-military types as mercs, for example), and another +1 if the hero has a detective skill/background). Note that the clue may not necessarily be a physical object; an informant with relevant dirt on the villain or organization is also applicable.
If neither a major encounter or major clue is rolled, then there is always a chance of a Minor Encounter. The odds of such an encounter, and the nature thereof, vary from neighborhood to neighborhood. Relative safe areas only have a 2-in-10 chance of a minor encounter, high-incident areas have a 4-in-10 chance, and most neighborhoods have a 3-in-10 chance of such an encounter (if for some reason the heroes head out into the suburbs, the chance of anything interesting drops to 1-in-20). This can lead to one of nine different types of encounter:
Assault - where one or more (non-powered) individuals are attacking one or more victims; this is often something as simple as a fistfight, although it can be as complicated as the GM wishes. There is a chance that an assault is actually a rumble, which involve two (or more) larger groups fighting each other; generally there is a 1-in-10 chance of this, but if street gangs operate in the area, it becomes a 1-in-6 chance. Note that a rumble does not have to include street gangs; rival sport fans, clashing racial groups, gays vs gay-bashers, and union members vs scabs are also possible rumbles, depending on the area in question.
Burglary - anything from a mugging to a late-night robbery to a bank heist.
Citizens Endangered - where normal people are put at risk, in such a manner that hopefully the hero can help to save the day. The situation might be either natural or unplanned (such as a bridge collapse or an apartment fire), or might be the product of human design (such as a kidnapping or hostage situation); the former is more likely in poorer areas, and the latter more likely in wealthier areas.
Contraband - where some illicit cargo is being moved from buyer to seller. Most likely drugs, but illegal weapons are also a popular option, and explosives, stolen goods, and even human cargo are possibilities.
Grand Theft Auto - where a vehicle or vehicles of some sort is being stolen. Optionally, this category also includes criminals using a vehicle of some sort as part of their crimes (a souped-up getaway car in a bank heist, or a helicopter from an industrial raid), which should ideally lead to a high-speed chase on the part of the hero.
Law Enforcement - an encounter with the local constables. If the hero is on good terms with the local authorities, this is usually not a problem, and may even result in the hero possibly getting some useful info as a result. If the hero is wanted by the law, or is disliked by the authorities, then the police may try to harass or arrest the hero. If there are NPC superheroes in the area, there is a chance that the PC runs into them instead (1-in-10 chance if only one NPC hero in the city, 1-in-6 chance if more than one, determine which one randomly).
Observation Rewarded - where the hero encounters some minor clue (perhaps through an informant or passerby witness) regarding a villain, organization, or street gang in the area, or a suspicious-acting person that might lead to the same. Either way, there's a 1-in-6 chance that it turns out to be a false alarm or red herring.
Vandalism - the destruction of private property (arson is a common example of this); this can range from spray-painting valuable works of art at a museum to planting bombs at a local sports stadium.
Weird Stuff - includes a variety of encounters that fall outside the usual superhero fare, which may or may not be immediately hostile or dangerous. This often has a supernatural element to it (ghost sightings, vampire attacks, etc), in which case there is a 1-in-10 chance that its actually a trick or scam of some sort. Not all weird stuff need be supernatural in origin; hearing voices from the future (created by a nearby high-tech experiment with tachyons) or extra-dimensional rifts temporarily showing up are non-supernatural examples of weird stuff.
Each neighborhood should have its own 'encounter table', listing both the chances of a minor encounter, and a table for what sort of encounter comes up. A baker's dozen of stereotypical neighborhoods are included below; modify for your individual cities and neighborhoods as you see fit:
RESIDENTIAL, WEALTHY (2-in-10 chance of encounter) 1 Assault 2-7 Burglary 8-9 Citizens Endangered 10 Contraband 11-12 Grand Theft Auto 13-16 Law Enforcement 17-18 Observation Rewarded 19 Vandalism 20 Weird Stuff RESIDENTIAL, MIDDLE-CLASS (3-in-10 chance of encounter) 1-3 Assault 4-7 Burglary 8-9 Citizens Endangered 10 Contraband 11-12 Grand Theft Auto 13-15 Law Enforcement 16-18 Observation Rewarded 19 Vandalism 20 Weird Stuff RESIDENTIAL, BOHEMIAN (2-in-10 chance of encounter) 1-2 Assault 3-7 Burglary 8-9 Citizens Endangered 10-12 Contraband 13-15 Grand Theft Auto 16-17 Law Enforcement 18 Observation Rewarded 19 Vandalism 20 Weird Stuff RESIDENTIAL, POOR (4-in-10 chance of encounter) 1-4 Assault 4-6 Burglary 7 Citizens Endangered 8-10 Contraband 11 Grand Theft Auto 12-13 Law Enforcement 14-16 Observation Rewarded 17-18 Vandalism 19-20 Weird Stuff RESIDENTIAL, SLUM (4-in-10 chance of encounter) 1-5 Assault 6 Burglary 7 Citizens Endangered 8-10 Contraband 11 Grand Theft Auto 12 Law Enforcement 13-16 Observation Rewarded 17-18 Vandalism 19-20 Weird Stuff CHINATOWN (2-in-10 chance of encounter) 1-2 Assault 3-4 Burglary 5-6 Citizens Endangered 7-11 Contraband 12 Grand Theft Auto 13-14 Law Enforcement 15-17 Observation Rewarded 18 Vandalism 19-20 Weird Stuff BARRIO (3-in-10 chance of encounter) 1-4 Assault 5-6 Burglary 7 Citizens Endangered 8-11 Contraband 12 Grand Theft Auto 13-14 Law Enforcement 15-16 Observation Rewarded 17-18 Vandalism 19-20 Weird Stuff CAMPUS DISTRICT (2-in-10 chance of encounter) 1-3 Assault 4-8 Burglary 9-10 Citizens Endangered 11 Contraband 12 Grand Theft Auto 13-16 Law Enforcement 17-18 Observation Rewarded 19 Vandalism 20 Weird Stuff WATERFRONT (3-in-10 chance of encounter) 1-3 Assault 4-5 Burglary 6 Citizens Endangered 7-12 Contraband 13-14 Grand Theft Auto 15-16 Law Enforcement 17-18 Observation Rewarded 19 Vandalism 20 Weird Stuff FINANCIAL DISTRICT (2-in-10 chance of encounter) 1-2 Assault 3-6 Burglary 7 Citizens Endangered 8 Contraband 9-10 Grand Theft Auto 11-14 Law Enforcement 15-17 Observation Rewarded 18-19 Vandalism 20 Weird Stuff INDUSTRIAL AREA, HIGH-TECH (2-in-10 chance of encounter) 1-2 Assault 3-7 Burglary 8 Citizens Endangered 9 Contraband 10-11 Grand Theft Auto 12-14 Law Enforcement 15-17 Observation Rewarded 18 Vandalism 10-20 Weird Stuff INDUSTRIAL AREA, BLUE-COLLAR (3-in-10 chance of encounter) 1-3 Assault 4-6 Burglary 7 Citizens Endangered 8 Contraband 9 Grand Theft Auto 10-12 Law Enforcement 13-16 Observation Rewarded 17-19 Vandalism 20 Weird Stuff INDUSTRIAL AREA, BETTER DAYS GONE BY (3-in-10 chance of encounter) 1-3 Assault 4-7 Burglary 8-9 Citizens Endangered 10-12 Contraband 13 Grand Theft Auto 14 Law Enforcement 15-17 Observation Rewarded 18-19 Vandalism 20 Weird Stuff(much of the info here on minor encounters is inspired by the DC Heroes supplement Night in Gotham, which took a similar approach to handling nightly patrols)
If no major encounters, major clues, or minor encounter are rolled for the time-block in question, then nothing of note occurs during that particular patrol. Even superheroes sometimes have slow nights...
(part three continues here)