One idea that I’ve seen floated about - but not by anyone who is going to actually build it- is a hybrid of the “private car on rails” and “Personal Rapid Transit” ideas- you have a overhead or underground (grade separated) rail system that has two kinds of cars on it. The first kind of car is public. You enter a station, and tell the station where you want to go. The station will pull the next available empty car into the station, and you enter the car. The car accelerates up to speed, and merges into the traffic on the rail - the traffic on the main rail continues past the station freely. (During heavy traffic periods, the traffic on the main rail might be slowed down or sped up to “make a hole” for the oncoming car.) You get to the destination, the car moves to the siding, and slows down and stops, you get out, and the car is either used by someone who needs it, or is sent to someplace where it is needed.
The other kind of car is a private car, which can be used on and off the rails. The driver of the vehicle enters it in his driveway, and drives it to an on ramp. Control of the vehicle transitions to the central system while it is on the rail system. The driver programs his destination into his vehicle, and sits back and enjoys the ride. When the destination is reached, the vehicle merges to an off ramp, and control passes back to the driver, who continues to his ultimate destination. Of course, there are safety concerns - the owner/renter of the private car would have to maintain certification of his vehicle before it could be used on the rails, which I would presume would require a combination of electronic monitoring of the vehicle’s health, periodical inspections, and consent to being kicked off the network at the first hint of a problem. Of course, the off-ramps would also need to be designed so that if the driver did not take control of the vehicle, it would not cause an accident.
You could design the system to allow for different types of vehicles to operate on it- including unattended cargo pods (can’t leave the rails, cargo stations?), attended cargo pods (replacement for delivery trucks), variety of personal pods.
It could be used for both inter-urban transit and intra-urban transit, depending on the network you design. Of course, you would need to have higher rail speeds on intra-urban transit, but you could design a system that would take care of both. Of course, for intra-urban, you would probably end up with private vehicles and private for-rent vehicles and not public vehicles. (I’d imagine that you could either rent a pod like you rent a car now, or rent a seat on a pod that traveled between two points, like an airline.)
For safety, you’d probably want both a central Pod Control system and an in-pod safety system (checks track ahead for obstacles, ability to follow a different pod and automatically brake to avoid or minimize collisions, - stuff that we can do with current technology).