If you hire a contractor to work on your home what insurance should the contractor have?

They should at a minimum have a commercial general liability policy for the protection of your property.

The contractor should also carry the necessary medical Insurance for his workers. The home owner is "not" responsible for injury to a contractor's employees unless the home owner was somehow personally responsible for that injury. Just having hired the contractor does not create a liability for the health and safety of the contractors workers.

The contractor is responsible for obtaining the appropriate protection for his workers as well as for his liabilities. Employee injury coverage is the responsibility of the contractor.

Answer

They should also definitely have Workman's Compensation insurance. If they don't and one of their employees gets injured while working on your project you could potentially get sued by the employee if the homeowner personally caused the injury. Again, make sure to see a copy of the policy.