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The traps were found in parkland and paddocks in Opoho by council contractors, council parks manager Lisa Wheeler said.
It was ''not acceptable'' for the traps to be laid in parkland and so close to walking tracks, she said.
''It's one of those things we don't want brought to our attention,'' she said.
''But there was an occurrence of people trying to catch possums as a way to get income and we were able to curb that last year and we certainly don't want that happening again.''
Each of the traps was surrounded by what appeared to be a mixture of white flour and aniseed to attract possums, she said.
The traps could cause damage to any animal or person who was caught by them.
''They would be big enough to get someone's leg or foot caught in them,'' she said.
While it was not illegal to use such traps, council contractors were not allowed to use them and ''we don't allow people to be trapping on public land''.
''We want to make the public reserves as safe as possible for people to use them.''
The traps were taken to the Dunedin Central Police Station and ''if anyone wants to try and claim them, they are more than welcome to claim them there'', she said.
SPCA Otago executive officer Sophie McSkimming said the discovery of the traps was ''really, really concerning''.
Some traps were found in a field with sheep on SPCA property.
While the traps were legal, they were a concern for pet owners and parents, she said.
''They don't do near as much damage as the old ones that had teeth, but they still do damage, especially if an animal is ... in them for days,'' she said.
She asked for people to think twice before laying traps in the city.
''It's really, really unsafe doing it [laying traps] in a built-up area and where there's so many animals.''
Sergeant Nathan White said police were trying to find out who the traps belonged to.
''They are somebody's property, so we are trying to track down who owns them and go from there,'' he said.
Neither the traps nor their use was illegal and they posed little risk to the public, he said.