In the first case, a car was found on fire just after midnight today in Locharbriggs.
Fire crews rushed to Wallamhill Road at 12.20am to tackle a burning red Vauxhall Corsa. They put out the flames but the car was seriously damaged.
After a joint investigation by the emergency services, officers say they believe it was a deliberate act.
Constable Paul Murphy, at Dumfries, said: “Extensive enquiries are being made into this reckless act and we are keen to trace a man who was seen in the vicinity the time of the fire.
“He is described as being white, in his early to mid 30s, stocky build, short dark hair and was wearing a black hooded tracksuit top with a black t-shirt underneath, black tracksuit bottoms and trainers. This man was walking three dogs, an alsatian, a black labrador and a small white dog.”
Meanwhile, a door and letter box of a flat in Kirkcudbright were damaged by fire yesterday afternoon.
Neighbours of the property in Dee Road put out the flames but called fire and police, who examined it and said it appears to have been deliberately set on fire.
Constable Gillian Dingwall, of Police Scotland at Kirkcudbright, said: “The consequences of such a reckless act don’t bear thinking about in this case.
“We want to hear from anyone who may have seen anyone acting suspiciously around the Dee Road area, and in particular in the vicinity of the flats around 3 pm yesterday.”
Anyone with any information which might help in either case should call police on 101.