If the repair man yjat my landlord hired to replace my range ripped a hole in my couc, who is responsible and what legal action do I take to resolve it?

An employer is liable for the negligence of an employee which occurs during the course and scope of employment.

If the repairman is an employee of the landlord, the landlord is liable for the repairman's negligence.  You would name both the landlord and the repairman as defendants in your lawsuit for negligence.

If the repairman is NOT an employee of the landlord, but is an independent contractor, the landlord is not liable.  You would name only the repairman as a defendant in your lawsuit for negligence.  If the repairman is an employee of a company, you would name the company and the repairman as defendants in your lawsuit for negligence.

Negligence is the failure to exercise due care (that degree of care that a reasonable repairman would have exercised under the same or similar circumstances to prevent foreseeable harm).

Your damages (the amount of compensation you are seeking in your lawsuit for negligence) would be the cost of repairs to the couch.  If the couch cannot be repaired, your damages would be the cost of replacing the couch.  You will need to mitigate (minimize) damages by selecting a replacement couch comparable to the damaged couch.  If you were to select the most expensive couch you could find, you would have failed to mitigate damages and your damages would be reduced accordingly.

You can file your lawsuit in Small Claims Court.  Your damages would include the cost of repair or replacement of the couch and court costs.  Court costs would include the court filing fee and process server fee.