You're riding in your city and you want to get from one street to another. If you were in a car you would need to stay on the public street and pay attention to the rules of the road, stopping at lights etc. But on the bike you can cut across a parking lot through an ally and on a sidewalk, as long as pedestrians are not in the way. Will anyone care? Most likely no. Some cities have rules against riding on sidewalks but if there aren't such laws a cyclist is welcome. There are no federal laws and rarely state laws governing sidewalks. Its up to the individual town or city.
Private property parking lots are also open to cyclists just as they are open to pedestrian walking through. There is really no issue cutting a corner through a private lot unless there is a no trespassing sign or a gate. If these are public places where customers are able to come and go such as a furniture store parking lot, then access is not an issue. Liability issues such as negligent maintenance are enforced if a cyclist gets injured but acts of weather such as ice or snow are typically not.
So what about cutting through a private driveway to access a public place on the other side? Suppose you want to ride across a property to acess some trails on a public park land on the other side. This is a different story. Business parking lots are public places. Even company employee parking lots allow visitors etc so crossing them is not a big issue but a private driveway, especially a home owners driveway, can be considered trespassing, especially if the owner tells you so. Different liability and privacy issues exist and cutting these properties is not going to be greeted with indifference. You can always ask for permission to cut through.
How about an empty house property, like one that is foreclosed and owned by a bank? Its still considered trespassing but enforcement would need to be first explained to the rider, either vocally or through signs. The owner or property manager would need to give a warning "Hey this is private property, don't ride through here!" or a properly spaced no trespassing sign. One solution would be to ask for permission here too. If you see a realitor or homeowner just say "Nice house. How much are you asking? blah blah. Hey you don't mind if I cut through the side here to get to the park do you? I can keep an eye on the place for you." The response might be positive and they know you're not some criminal, just a biker looking for a ride.