Police made the arrests after the fight broke out in Frodingham Road and Percival Street in the town, on Sunday afternoon.
Some of those arrested were alleged to have had makeshift weapons. Police said there was no suggestion that race was a factor in the disturbance.
Police said a small number of men sustained injuries and were taken to hospital, including one man who suffered a fractured eye socket and another who sustained a fracture to the hand and arm.
The men, who were arrested for violent disorder or possession of an offensive weapon, remained in custody yesterday at Scunthorpe or Grimsby Police Station.
A police spokeswoman said: “Following the prompt action taken by police there were no further incidents of disorder reported.
“At this time there is nothing to suggest any racial motivation to the incident and those arrested were all from the same community.”
The alarm was raised by a member of the public at 3.44pm.
Police said they made their first two arrests were made “within minutes” of their arrival on scene.
A short time later 13 more men were arrested with the final three were in custody by 4.12pm.
Humberside Police said “numerous” people contacted them to report the disturbance and they had spoken to many of the witnesses, but were keen to speak to anyone else with information.
They can be contacted on the non emergency number 101 and should refer to log 484.