Windows were damaged at three locations in the town through the use of an automatic-type catapult or air rifle, police say.
Officers were first called to the Morrison's store on Mill Street at 2.35am on Friday after reports of nine windows being smashed.
The force found significant damage to the 15ft-high windows – valued at around £20,000, with ball bearings being found at the scene.
Officers were called again at 7am on Sunday to Stone Leisure Centre where a total of 25 windows were damaged.
At around 8am on the same day they were called to the nearby M&S food store, where around 15 panels had been damaged. It is believed that the damage to the leisure centre and M&S took place between 3am and 3.15am on Sunday.
Officers are currently reviewing CCTV footage of the area at the time and ball bearings from the scenes are being forensically examined.
Chief Inspector Giles Parsons, of the Stafford Borough Neighbourhood Policing Team in Stone, said: "We are treating these incidents extremely seriously.
"I am particularly concerned about someone firing ball bearings, causing significant damage to property and the potential that these have to possibly cause someone serious injury.
"I also recognise the impact these incidents are having on businesses. Officers would therefore like to speak to anyone who was out in Stone town centre in the early hours last Friday or Sunday.
"We are particularly keen to hear from two women who were walking across the bridge on Stafford Road adjacent to the Canoe Club at approximately 3am on Sunday who we believe may have encountered the individuals we suspect as being involved."
Any friends or relatives of anyone who has an interest in, or owns a catapult or other device capable of causing this type of damage, are also urged to contact police.
Chief Inspector Parsons added: " "I would like to reassure the local community that we are thoroughly investigating these matters to try and catch the persons responsible and will be further increasing our presence in the area."
Anyone with information is urged to contact Staffordshire Police on Facebook, Twitter, or call 101, quoting incident number 050 of June 25.
Alternatively, contact Crimestoppers anonymously on 0800 555111 or at crimestoppers-uk.org