Police search dump for shoe in murder case

The needle is a sneaker and the haystack is a rubbish dump.

Canberra police are sifting through tonnes of stinky trash in search of a single shoe which they believe holds vital forensic evidence in the New Year's Day murder of 71-year-old Miodrag Gajic.

"We're searching the tip for a shoe we believe was worn by the offender during the incident," Detective Sergeant Anthony Brown told reporters.

Described only as a "Puma trainer", police refuse to reveal what lead them to the Mugga Lane rubbish tip, or the whereabouts of the matching runner.

More than a month after the killing, Sgt Brown admitted officers and SES workers face a difficult task.

"Obviously it's a laborious task that we've got to get through but we'll be here for as long as we need to," he said.

Temperatures in the nation's capital are forecast to reach 37 degrees in coming days.

"As it does warm up it's going to get tougher. It is hard work and it is painstaking."

Mr Gajic was found dead in his south Canberra home on January 1, with investigators confirming the grandfather died from head injuries.

Police have identified a suspect who they say was known to Mr Gajic.

Media reports say the suspect is in custody for other matters.