The answer
Booming/Banging in furnace: delayed ignition (gas buildup before ignition; can crack heat exchanger). Popping ductwork: thermal expansion and static pressure flexing. Squealing: failing motor bearings or fan belt. Buzzing: bad contactor relay or seized compressor. Clicking: faulty ignition relay or flame sensor.
Sources
This answer is drawn from the standards and guidance below. It is general information about how these systems work — not a diagnosis of your property.