The incidents have been reported in the village of Whittington near Oswestry.
Those responsible have targeted a small area of the village, officers said.
Oswestry's Safer Neighbourhood team said such behaviour could cause serious damage to life and property.
The officers tweeted: "There have been several reports of lit cigarette ends being put through letter boxes on Boot Street and the Penybryn areas of Whittington.
"Obviously the outcome of this behaviour has potential to cause serious damage to life and property."
West Mercia Police is appealing to witnesses or anyone who may have information about the incidents to contact them on 101 or use Crimestoppers, 0800 555111.