One woman used the public site to report her teenage son missing while others asked if officers had a hovercraft and what time a car boot sale opened.
It has prompted Northamptonshire Police, which has been using the popular social networking website to highlight operations and appeal for witnesses for the past two years, to tell residents to use the traditional method of the telephone instead if they want to report a crime or anti-social behaviour on their streets.
The force's corporate communications team, which monitors the site, wrote online on Monday: "Morning all, we've had lots of people reporting crimes over the weekend on Facebook - unfortunately we are unable to monitor the page 24/7 so if you have an issue please call us on 101 or in an emergency (crime in action, fear for safety for example) then always dial 999."
Northamptonshire police's Facebook page, which is "liked" by 18,971 people, has a wall on which anyone can leave public messages. It removes posts if they name alleged offenders or include offensive language.
But many messages remaining on the site include sensitive information about supposed crimes or victims, as well as apparent hoaxes.
The most recent wall post claimed that someone was driving a car while sitting on a sofa on its roof, while another commenter claimed that a road had been blocked off so Superman could retrieve a lost sock from a drain.
One woman, whose name and detailed Facebook profile was visible to all, reported that her 13-year-old son had gone missing after an argument.
Another named a local man and said she had seen him looking "really rough" with black eyes in Northampton town centre.
Others reported men hanging around suspiciously outside skateboard parks and blocks of flats.
Many messages were queries from residents asking why particular roads were closed or why police helicopters were over their streets, as well as reports of dangerous driving.
One man asked if the East Midlands constabulary had a hovercraft, as it would be "awesome" if they did.
The force replied in a comment that it did not, but added that it was able to borrow a tractor from Thames Valley Police.
It is the latest example of the openness of the internet causing problems for police.
Earlier this year Strathclyde Police announced the arrest of Andy Coulson, the former Downing Street communications chief and News of the World editor, on its Facebook page.
But because of the design of the site, it was unable to stop members of the public clicking on a button to say they "liked" the development and adding comments about it.
Sussex Police hired new press officers to use social media such as Facebook and Twitter, but one of them posted a photograph of a desk covered in cakes along with the message "a typical Friday in coroporate comms".