Hazel North, 19, from the Cairneyhill area of Dunfermline, last contacted her sister on March 4 and told her she was travelling to meet a friend in Kilmarnock.
She has not been in touch with her family since, police said.
Forensic teams have been searching a flat and garden in Northcraig Road, Kilmarnock, and officers carried out door-to-door inquiries.
Nothing untoward was found and officers are "following a number of lines of inquiry", including a possible new sighting of Ms North in Coronation Drive, Stranraer, on Tuesday March 18.
Ms North is 5ft 8in, with a slim build, a fair complexion and shoulder-length blonde hair. Police said she used to live in Stranraer.
The potential sighting of her was at around midday on Tuesday and she is thought to have been in the company of another woman of similar height with a medium build and short, dark hair in a bob style.
She was wearing a dark coloured hooded top and dark blue jogging trousers.
Police are continuing investigations in Dunfermline, Kilmarnock and Stranraer and particularly want to speak to the woman who may have been seen with Ms North last week.
Superintendent Neil Kerr said: "What I can tell you is we've carried out an extensive search of the house and the rear garden, as we would do with any missing person inquiry, but I can tell you at this time that we haven't uncovered anything that would give us any cause for concern at this stage."
He added: "We are currently following a number of lines of inquiry, speaking to family and friends to ascertain more information about Hazel and as to where she may be.
"Her family is obviously concerned by the lack of contact from her and they just want to know she is safe and well. If anyone has any information which could help us locate Hazel, please do contact us."
Anyone with information can contact police on 101.