Bradley's friends and former co-stars have paid tribute to the late actor, including the author of Trainspotting, Irvine Welsh, who wrote: "Bradley John Welsh, my heart is broken".
Police are treating the death as suspicious and are asking witnesses to contact them with any relevant information.
"I then went to prison for extortion and firearms and all that stuff", he said at the time.
He appeared in Trainspotting 2 alongside Ewan McGregor, Johnny Lee Miller, Robert Carlyle and Ewen Bremner, playing the gang figure Mr Doyle.
A fifth wrote: "When you're out the country and not reading news properly from home, and think Bradley WALSH has been shot in Edinburgh, and even worse, you've told a couple of fellow Brits that the presenter of The Chase has been shot".
Scotland's national newspaper reported a neighbourhood resident saying that the gunshot wound was to Welsh's head.
"We've not been told anything but police waved through some ambulances".
Armed police were called to the scene following reports of a disturbance and the street was cordoned off.
Trainspotting 2 star Bradley Welsh has been killed in a gun-related incident in Scotland. Goodbye my wonderful and lovely friend.
He added: "Thanks for making me a better person and helping me to see the world in a kinder and wiser way".
Bradley Welsh was also a prominent member of his community, owning his own boxing gym in the Edinburgh area of Holyrood, where he endeavoured to keep young people away from crime through various charity projects. "Thoughts are with his family and friends".
He said: "Devastating news about Brad Welsh tonight, a huge character in Scottish amateur boxing and the Hibernian support and someone who contributed a great deal to society through his charitable work and boxing gym".
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