The 36-year-old has not been seen since leaving the Dakota Hotel in South Queensferry around 1am yesterday morning.
Police at a press conference earlier this afternoon said that they are particularly keen to speak to two people who may have seen Scott in the vicinity of the bridge.
Police Scotland said Mr Hutchison had left the hotel on foot and that they were “keeping an open mind” as to his direction of travel.
They said particular focus was being given to the South Queensferry area, Forth Road Bridge and Fife.
Police said CCTV analysis had identified two people seen crossing the footpath of the bridge around 1.10am and urged them to come forward.
Chief Inspector Alan Carson said: “It is important to stress that at this time we do not know definitively where Scott has headed after leaving the Dakota Hotel and the assistance of the public is therefore vital to help us with this investigation.
“Anyone who believes they have seen Scott since 1am on Wednesday morning should get in touch.
“In particular, the two people who crossed the Forth Road Bridge from South Queensferry at 1.10am may have useful information and we are extremely keen to speak with them.
“Finally, I’d like to reiterate our appeal to Scott directly, to make contact with family, friends, or with police and let us know where you are.”
Messages of hope were posted on social media from friends, family and fellow musicians, including Gary Lightbody from Snow Patrol.
There was also support from authors Irvine Welsh and Christopher Brookmyre, and radio presenter Edith Bowman.
His brother and bandmate Grant Hutchison said the family felt helpless but that the support they have received had been incredible.
The singer is described as white, 6ft tall with a stocky build, dark hair and a thick beard.
He was last seen wearing a dark baseball cap, navy blue hooded jacket, grey or khaki trousers and white trainers.
Anyone with information is urged to call police on 101.