The attacks began near Broomhouse Mosque in the west of Edinburgh at about 20:30 on Friday.
Two 22-year-old men, understood to have recently left Asr prayer, were stabbed multiple times at Sighthill Park and taken to Edinburgh Royal Infirmary.
A taxi was then vandalised at a Shell garage on Telford Road, about five miles away, at 21:15, leaving it with broken windows.
An image from the scene appeared to show a hatchet with a yellow and black handle lying in the back seat.
CCTV footage showing a man standing beside a black vehicle with its window smashed at a BP garage on Ferry Road was recorded at 21:28.
The video shows the man entering the kiosk before overturning shelves and leaving products scattered across the floor.
