Alathea Byrne – known as Alice – was last seen by her family at around 2am on Saturday, January 1.
It is understood she was heading to a beach party to celebrate the New Year.
She later met friends in Marlborough Street and left a flat there at around 10am on the Saturday.
Police are now appealing for help to trace the 28-year-old as concerns grow for her safety.
Inspector Keith Scott said: “It is out of character for Alice not to keep in contact with her family and friends. We are becoming increasingly concerned for her wellbeing and we are working to trace her as soon as possible to ensure that she is safe and well.
"Searches and enquiries have been ongoing since she was reported missing, and we are now asking the public for any assistance they can provide.
“I would urge anyone who has seen Alice since Sunday morning, or anyone with information on her whereabouts to contact us as soon as possible.”
Ms Byrne has been described as white, around 5ft 6in tall, medium sized with short black hair.
When she was last seen she was wearing an orange jacket, black top, black jeans and white trainers. She was carrying a black rucksack.
If you have any information on Ms Byrne’s whereabouts you should contact Police Scotland on 101 quoting incident number 0647 of January 2.