For installed carpet you want a carpet with cut pile and a heavy density. The taller the pile the better. You want foam underlay with a density of 8lb or higher. For rugs, a dense heavy rug covering as much of the front of the room as possible.
Thin, rubber backed carpets are not as good (as found in commercial offices) but wear longer in high traffic areas. Their acoustic absorption coefficients can be found online. I would recommend to go to a flooring store and price several options if getting installed.
Adding a huge rug or carpet to an existing room will drop decay times by 40-60%. The larger the room the better the improvement. Bed bath beyond has a bunch of huge rugs that are attractive, thick, and can be taken with you when you move down the road.
