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The most concerning case involved a father driving at almost three times the legal limit with his child in the car.
Senior Sergeant Craig Dinnissen said the man was stopped on Saturday at a police checkpoint at about midnight in the central city.
He blew a reading of more than 700 micrograms of alcohol per litre of breath.
Also in the car was his 10-year-old daughter.
Police now had serious concerns for the welfare of the child after the incident, Snr Sgt Dinnissen said.
Officers were also called to a car crash in the suburb of Calton Hill.
A 33-year-old was arrested after crashing his car into a power pole.
At first he fled the scene, only to return later.
It then emerged he was subject to a zero-alcohol licence and when tested he blew more than four times the legal limit, Snr Sgt Dinnissen said.
A 16-year-old was also pulled over past their curfew late on Saturday and blew a breath-alcohol reading of 390 micrograms - drivers under the age of 20 could not consume any alcohol.
Mr Dinnissen said the incidents were a concern for officers in the city and drink-drivers would continue to be targeted.