Residents say the damage has left a hole in the middle of the road outside the former Achurch and Sons premises.
They claim it is a danger to pedestrians, cyclists and drivers.
Former town councillor Richard Barker is one of several residents who contacted News to say they had reported the issue via the County Council portal.
Mr Barker said: “We’re told to report any issues via the proper channel.
“I contacted the County Council more than three weeks ago and yet nothing had been done. I’m sorry, but that is not good enough.
“I appreciate it is not a main road but it is still very busy. There are a lot of pedestrians using this road - many of them are elderly.
“Someone could easily trip in this hole, especially at night because the area is not very well lit.
“If someone was injured, perhaps the council might do something about it.”
Mr Barker said he had raised the matter with the town’s county councillor Bill Aron.
Liam Collins revealed he had also reported the damaged cobbles.
He added: “I drive down this road at least three or four times a day and every time you hit that hole, there’s a clunk under the car.”
Mr Collins said the longer the hole went unrepaired, the larger it would become - adding to safety concerns.
He added: “I know the cobbles are a part of the town’s history but a lot of them are in a bad state. Perhaps they should dig the lot up.”
His comments follow complaints about the standard of repairs in another area of the Market Place.
Contractors dug up York stone flagstones - only to replace them with tarmac.
However, Richard Davies - Executive Councillor for Highways - says the County Council is not obliged to replace materials like-for -like.
He told town councillors recently: “We don’t have to comply with rules regarding historic materials.
“Our statutory duty is to maintain highways and keep them safe.
“I know I won’t be popular but I am increasingly concerned by the use of heritage materials. They fail and become dangerous very quickly. That is why we give consideration in favour of using the hardest wearing materials.
“We’ll try to be flexible. York stone flags move and require huge amount of investment. They are not always best use of council taxpayers’ money.”
The News contacted Lincolnshire County Council about the damaged cobbles.
The authority pledged the cobbles would be repaired ‘in the very near future’ but did not give a date when the work would be carried out.
In a statement, the local highways manager said: “We’re aware of the issue with the cobbles in Market Place, and they should be repaired in the very near future.
“We realise situations like this can be frustrating for people and we apologise for any inconvenience caused.
“We thank people for their patience.”