X sales vehicle to Y.
Theres a bill of sale to prove it.
X drops the insurance on vehicle.
Y then has an accident, but the license plates that had originally belonged to X had not been taken off yet, and Y has not retained new insurance yet.
Who is responsible for the damages done?
It all depends on the situation. If you bought from a dealer than you would have been given a temporary 30-day transit tag for the registration. Insurance is really dependent on your carrier, but most of them have a rider for just purchased vehicles that will cover you for a certain period of time automatically. This time can range from 24 or 48 hours to 30 days.
If you bought the car off a private individual and you did not have anything on it, then you shouldn't have been driving the car.
Your facts are incomplete as you don't indicate who caused the accident. Under Missouri law the only party liable is the party (or parties) whose negligence resulted in the damage. Generally speaking that means the driver. If Y was driving the vehicle and his/her negligent operation of the vehicle caused damage then Y is liable for the damages. The rest of your hypothetical seems to be centered around who was the owner of the vehicle at the time of the MVA which generally has no bearing on the liability issue as it isn't ownership of the vehicle that caused damages but rather operation of the vehicle.
That said, it is possible to hold the owner of a vehicle liable for damages caused by another's operation of the vehicle under a theory of negligent entrustment. Essentially this theory holds that I let Mad Dog Joe drive my car knowing that he has been in 42 accidents with 28 convictions for reckless driving. Again, mere ownership does not create any liability on my part but rather my negligence in entrusting the vehicle to someone that I knew was likely to operate it negligently. I've oversimplified the theory but its important to understand that generally this is a very difficult case to make. Under the facts you cited I would think that it would be near impossible to establish negligent entrustment as Y would have the legal right to operate the car and X would have no legal right to prohibit him from doing so. While I can be held liable for negligent entrustment by permitting someone else to operate a vehicle I own there is no such thing as negligent selling the vehicle to someone even if I believe the new owner is likely to drive the vehicle negligently.
THIS IS NOT LEGAL ADVICE - NO ATTORNEY-CLIENT PRIVILEGE ATTACHES - FOR INFORMATION ONLY. DO NOT RELY ON ANY ADVICE YOU RECEIVE FROM ME OR ANY OTHER ATTORNEY IN THIS FORUM. Legal advice ONLY comes after a complete review of the facts and relevant documents and an expressed (written) agreement of representation that forms attorney-client confidentiality. Neither of these two events can occur in this forum. Answers to any Avvo question are rooted in general legal principles - NOT your state's specific laws. There is no implied or actual attorney-client relationship arising from this educational exchange. You should speak with an attorney licensed in your state, to whom you have provided all the facts before you take steps that may impact your legal rights.
Insurance and vehicle titles have nothing to do with liability. The tortfeasor is the person responsible for the accident. That said, insurance coverage may be another issue. Insurance follows the named insured and named vehicle. This means, if X had insurance on the vehicle, there may be some recoverable insurance proceeds, assuming an exclusion does not apply. This is often the case when a permissive driver is operating someone else's vehicle without insurance. Although the owner wasn't the tortfeasor, his insurance may cover the damages. This situation is called the "omnibus clause" whereby an insurance policy extends coverage to any person using or responsible for the use of the insured vehicle.
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