The first attack was carried out by a female suicide bomber at the Baga fish market at roughly 11:20 am (local time), said Abubakar Gamandi, the head of the fisherman's union in Borno State, of which Maiduguri is the capital.
"A female suicide bomber exploded as soon as she stepped out of a motorised rickshaw," said Gamandi, who was at the scene. "Eighteen people were killed."
That account was supported by a nurse at the Augury General Hospital, where many of the victims were taken.
It was not immediately clear whether the second attack was carried out by a suicide bomber.
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"For now, we have at least 33 dead bodies from the two incidents...The toll may change because we are expecting more casualties," Gamandi told AFP.
The first two attacks were confirmed by Borno's Justice Commissioner Kaka Shehu, but he declined to provide casualties figures until rescue workers had completed their searches.
Shortly after 1:00 pm (local time) Shehu phoned AFP in Maiduguri to confirm a third bombing at the busy Borno Express bus terminal, but details, including on casualties, were not immediately available.
"The terrorists are angry with the way they were sacked from towns and villages and are now venting their anger," he added.
Shehu was referring to an ongoing offensive against Boko Haram in Borno state being waged by Nigeria and Chad, with support from Cameroon and Niger.
The four-nation offensive has claimed major successes over the insurgents since the start of last month, and analysts have said that Boko Haram would likely step up bombings on civilian targets in response.