Officers can - and often do - run plates of cars that have not committed any moving violations. A police officer has the legal right to act upon any information he lawfully obtains. When someone drives on a public road, they display their license plate at that time. A police officer who observes the plate can run it. If the plate comes back as resistration expired, the officer then has probable cause to stop the car and cite the driver.
DISCLAIMER: This posting is not intended to provide legal services or legal advice and it does not create an attorney-client relationship.
I agree with my colleague. In addition to the assessments in his his posting, the police can simply run plates randomly, such as every 3rd, 5th, etc, vehicle. What the discover as a result allows them to take further police action. Good luck.
DISCLAIMER I do not practice law in your State. This answer is provided solely for informational purposes only. This answer does not constitute legal advice, create an attorney-client relationship, or constitute attorney advertising.
Yes, they can and do. Not only can they randomly check plates, it is their job. They are looking for any potential violations from uninsured vehicles, unregistered vehicles, to people with outstanding warrants for a variety of reasons from the courts.
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Several Texas police departments are either currently or have announced plans to purchase license plate scanners for use on patrol cars. These scanners automatically scan the license plates to check registration information and can conceivably be sued to run criminal histories of their owners. Police officers plan to use this information to detain the drivers of those vehicles. As the other attorneys have pointed out, you don't have an expectation of privacy is something that is displayed to the public, so if they can see it (and they have to be able to see the LP for you to legally operate the vehicle), they can run it.
Disclaimer: This answer is provided as a public service and as a general response to a general question, it is not meant, and should not be relied upon as specific legal advice, nor does it create an attorney-client relationship.