The dead four year-old ewe was discovered on farmland on the outskirks of the town on Friday.
The farmer who came across the slain animal found it had sustained a fatal head wound.
And a postmortem examination suggested that it had most likely been shot with a crossbow and arrow.
Officers investigating have urged anyone with information to come forward.
The incident has left people across the region shocked and unsettled while the culprit remains at large.
Many have taken to social media to share their dismay.
Lisa Mitchell wrote: “There are no words to describe my feelings towards the monsters who got pleasure from inflicting horrific injuries on the poor sheep.”
Stef Spodie added: “It’s sick enough to just shoot some poor beastie without having to get your hands dirty just to feel like a bad man but an arrow has to be removed too! That’s not a normal person.”
And Frances Graham said the authorities should be “more vigilant” about who is able to obtain weapons such as crossbows and guns.
Furthermore, at Moffat and District’s monthly meeting on Tuesday night, chairman Leys Geddes called the incident “absolutely disgusting”.
A Police Scotland spokesperson said: “We are appealing for information following the discovery of a sheep with fatal injuries on farmland on the outskirts of Moffat on Friday 24 September 2021.
“The four-year-old ewe was found by the farmer with a head wound. Post mortem examination suggests that a crossbow and arrow is the most likely cause of the injury.”
And Sergeant Scott McDowall at Annan added: “Initially it was believed that a bullet has caused the wound/death, however on examination by a vet, no bullet was found in the skull nor was there an exit wound, therefore it would appear that it was a crossbow that was used to inflict the injury and cause the death.
“Please quote PDG0207930921 if you have any information on 101 or Crimestoppers on 0800 555 111.”
