They happened in Ballycastle and Armoy in the early hours of Saturday, June 22.
Three men arrested over the incidents have since been freed from police custody.
Detective Sergeant Rachel Gardiner said: “We believe at this time that an attempt was made to break into a shop at Main Street, Armoy, at around 12.50am on Saturday morning.
“After failing to gain entry those responsible then proceeded to a petrol station on the Coleraine Road in Ballycastle where entry was gained and a number of cigarettes taken.
“Sometime around 2am, Northern Ireland Fire and Rescue Service received a report that a car had crashed on the Loughareema Road in Cushendall.
“When they attended, the car was on fire but there were no persons at the scene.
“We believe this car, a dark coloured Vauxhall Insignia, was involved in both earlier incidents.
“Three men have been arrested in relation to our investigation and have since been released on police bail as our enquiries continue.
“I have a number of appeals for assistance from the public I would like to make.
“Did you see a dark coloured Insignia being driven in the Armoy/Ballycastle areas in the early hours of Saturday?
“Did you witness the collision at Loughareema Road involving this car?
“Did you see anyone walking between Loughareema Road and Armoy, or did you give a lift to anyone in this area?
“Do you have any dashcam footage from any of these areas?
“If so, please call detectives at Coleraine on 101 quoting reference number 118 22/06/19.
“Alternatively, information can also be provided to the independent charity Crimestoppers on 0800 555 111 which is 100% anonymous and gives people the power to speak up and stop crime.”