You ask "...what should you check for..."
The answer is "...what ever you want it to be..."
The terms of the engagement contract should specify what the council requires the contractor to be insured against and to what level.
The only legal requirement for an employer is Employers' Liability insurance as defined by the Employers' Liability (Compulsory Insurance) Act 1969 and there are exemptions to that requirement if the employee is a close family relative. Even this is a requirement which rests with the contractor NOT the engaging entity.
There is an unfortunate and widely held 'belief' that contractors must hold public liability insurance. This is not a legal requirement but it has crept, seemingly irreversibly, into just about every facet of common practice.
The essential element that so often gets lost in the apparent hysteria is the consideration of what / who is actually "at risk" and who is liable should any mishap occur.
There is no requirement for a contractor to hold public liability insurance. That is not to say they hold no responsibility for repair, renumeration or compensation for reckless or careless action resulting in damage or harm. The liability will always rest with the person occasioning the damage / harm regardless of what insurance they may or may not hold. Similarly, there is no compulsion upon a contractor to register a claim against their insurance in the event of damage or harm.
It is perfectly reasonable for any entity seeking to engage a contractor to set and define the terms of the engagement. The point which is so often missed is that the greater the complexity in the terms of the engagement = the greater the cost of the service provision. Expenditure from public money must be subject to the greatest degree of scrutiny and management.
In the original question road side verges are mentioned. It may be a requirement of the local authority that work in or adjacent to the highway requires a 'licence' and that the person has suitable qualification for street works operative / supervisor. There are certain exemptions for 'mobile' works as defined in the Safety at Street Works and Road Works Code of Practice https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/321056/safety-at-streetworks.pdf
The terms of the licence may state that PL insurance of £10million is required.
The point remains - when a contract is let, the terms must dictate specific qualifying criteria.
What do you want / require your contractor to be insured against and why?
The higher the complexity of the engaging entities requirement as set out in the contract directly effects the cost.
