Emergency crews, including police officers, firefighters and paramedics, were called to Phoenix Bank at around 6.45pm.
They found a partly collapsed wall with the car a short distance away.
Shropshire Fire & Rescue Service said one crew from the town's fire station was called out after receiving reports of a car hitting a sandstone wall.
Market Drayton Fire Station posted photos from the scene.
In a statement, a spokesperson said: "On arrival at the scene the crew discovered that a vehicle had been involved in a significant collision with a large sandstone wall. The vehicle was located a short distance from where the collision had occurred.
"Officers from West Mercia Police were on scene and were liaising with the driver, to establish the details of the collision.
On inspection of the wall by the crew, it was established that an area of approximately two meters had been demolished and collapsed due to the collision. A further area of wall was found to be structurally unsafe.
"The crew established a safety cordon, cleared debris from the road, and requested the attendance of Shropshire Council, and the highways department to close the footpath and deal with the unsafe structure."
It's understood that a man was taken to hospital with arm injuries, while the firefighters remained at the scene for approximately one hour.
West Midlands Ambulance Service sent an ambulance an paramedic officer to the scene. A spokeswoman said: "The driver, a man, was given treatment on scene for non-life threatening injuries before being conveyed to Princess Royal Hospital for further care."
West Mercia Police also attended the scene.