The 36-year-old band member was last seen at around 1am on Wednesday after leaving the Dakota Hotel in South Queensferry.
Police Scotland discovered a body at nearby Port Edgar at 8.30pm on Thursday.
Formal identification has yet to take place, but Mr Hutchison's family has been informed.
Fellow musicians expressed their sadness while tributes flooded in from fans.
Franz Ferdinand singer Alex Kapranos tweeted: "Awful news about Scott Hutchison. A terrible loss."
Stuart Murdoch of Belle and Sebastian wrote: "Tragic news about Scott Hutchison. The whole music community in Scotland was praying for a different outcome.
"Folks, if you are up against it, having dark thoughts, please tell someone, family, a friend or a doctor. There is always another way, though it might not seem like it."
DJ Edith Bowman wrote: "Can't really believe I'm reading this. Saddest awakening ever. Love and best wishes to all the Hutchison and Frabbit family", and author Ian Rankin tweeted: "No no no. Hellish news."
More than 1,200 comments have been posted on the singer's own Twitter account, with one fan writing "You've left a legacy of beautiful music."
The police search was focused on the South Queensferry area, the Forth Road Bridge and Fife.
On Thursday Mr Hutchison's brothers, Grant and Neil, said they were "distraught" about his disappearance and appealed for him to get in touch.
The pair spoke about their sibling's fight with depression and how he had helped others by speaking about his condition.
Elder brother Neil, 38, said: "In relation to Frightened Rabbit, the band and Scott's work and how many people he himself has helped who might be suffering from the same emotions and feelings, that's been great to read."
He added: "He does suffer from depression, he has been very open about that in an attempt to help other people with similar conditions.
"He's kind, he's a great uncle to my three kids, a fantastic wee brother and an inspiring guy who is one of my heroes. "
"He's an inspiration to me and I miss him very much."
Grant Hutchison, 33, a drummer with the band, spoke about his brother being an "inspirational" figure and a "hero" who he missed very much.