That depends on how the council is approached.
I holed the sump on the Cossie in London on a piece of raised ironwork. Took pictures, got the car AA'd home, got three quotes and presented them to the council in question. Their first response was to try to blame contractors. I said no, they were the Principal, the contractors are a third party and of no interest to me. The final paragraph said they had 28 days to pay or I would see them in the small claims court. 10 days later I had a cheque with the whole process from bang to back on the road taking 6 weeks.
I made an effort to learn how to effectively go about making the claim successful and was lucky enough to get some very sage advice. The small claims court can be a powerful weapon, even if you never go anywhere near one. Be polite and reasonable and there is very little that can be done to prevent a successful action. Reasonable is key because these are civil issues and in a court the test of reasonable-ness is applied rather than beyond reasonable doubt. If you are reasonable I'll bet 10-1 that at least once council bod won't be and from then on, they lose. Or put another way, give them enough rope to hang themselves... Then the amount by which they lose is really up to them. And be prepared to strike a deal. For example, offer to waive some out-of-pocket expenses or perhaps offer to pay an equivalent of your own insurance excess as a gesture of good will in return for an early settlement. Tell them the offer is time limited and once the deadline passes you will claim the full amount. I've won more than the case above in this fashion so believe me when I say this approach really works. They may not care about potholes or whatever but they do care about their bottom line.