The incident happened in Riverside Road between 9pm on Thursday, October 5 and 8am on Friday, October 6.
Between these times, a rabbit hutch situated in the garden of a property was opened.
Two guinea pigs have been reported as stolen and a rabbit removed and killed. A veterinary examination suggested that the rabbit had been killed by a human.
PC Graham Green, who is investigating, said: “This was understandably an extremely distressing incident for the rabbit’s owner and I would like to reassure the community that we are doing all we can to locate the offender.
“While taking on board the guidance from the vet, we are keeping an open mind surrounding the offender as a fox was seen in the area around the time of the incident. I am appealing for anyone who saw or heard any suspicious activity in the area between the times stated to please get in touch.
“My advice to pet owners would be if you are concerned, there are steps you can take such as moving your animals to a locked shed or outbuilding at night time if possible, or improving the locking mechanisms on their hutches.”