This is where Madison differs from a lot of towns. The City's contracted curbside service covers household trash and recycling, picked up twice and once a week, but it will not take construction debris, carpeting, or building materials. Those have to leave the property another way.
Most of it goes to the Canton Landfill, run by the City of Canton roughly 12 miles north of Madison. The landfill office is at 303 Soldier Colony Road, Canton, MS 39046, and the number is 601-859-8626. It accepts construction and demolition debris by name, along with household garbage, natural vegetation, tires, and white goods.
The service area matters at the gate. The City of Canton takes waste from Madison, Hinds, Holmes, Simpson, Warren, Copiah, Rankin, and Yazoo counties, so a Madison job qualifies. The city publishes its rates as a per-ton charge plus an environmental charge and a fuel surcharge, and it does not post operating hours, so call ahead if you plan to haul anything yourself.
Your account manager will confirm which facility your load is going to and what the charges will be before your container is delivered.