Police are investigating the incident which happened sometime during Saturday night and Sunday morning.
A spokesman at Dumfries said that the vehicle targeted, a blue Ford Focus, disappeared overnight from the driveway of a house in the Summerpark area.
It was later discovered in the Lochfield Road area at about 7.35 am on Sunday.
The fire was extinguished by the Fire and Rescue Service, but not before it was extensively damaged.
PC Chris Kemp said: “We are keen to trace the following described man who was seen in the Callum Road area at around 3.45 am on Sunday.
“He appeared to be ringing doorbells in the street and is described as being in his late 20s, had a beard and was wearing a black jacket.”
PC Kemp added: “We are also keen to hear from anyone who may have seen this car being driven in the Dumfries area in the early hours of Sunday morning.
“The number of the car is SD 10 XEU.”
Anyone who has any information which might help is asked to call police at Dumfries on 101.