There is a lot of emphases put on insulation. It’s very intuitive that insulation would add a great deal of performance in a wall or ceiling. Lab data tells us otherwise. Insulation helps and is certainly an audible improvement over an empty wall or ceiling cavity. However, insulation does less than added mass, damping, or decoupling. So while intuitive that insulation would do a lot, it does not in reality.
It is also intuitive that different insulations would perform differently, or that exotic insulations would work better than plain old fiberglass batt insulation. The fact is that fiber insulations work very similarly. They’re all similar density fibers, after all. One thing to note- foam is not what we want at all. It could be blue foam billets, open-cell, or closed cell. All foams are too dense and will actually conduct a vibration.
If you have a deep wall cavity or a double stud wall you might consider adding 1” of foam to seal things up nice, then fill the balance of the cavity with fiber. Best of both worlds.
Otherwise, I’d suggest getting the least expensive fiberglass insulation. Fiberglass tests best in the lowest frequencies, and at the end of the day, the low frequencies are generally all that remain as audible. I would recommend fiberglass over Mineral Wool insulation.
Mineral Wool has a fire rating however and can be used as fire-stop instead of rigid drywall. That’s an example of when you might use both types of insulation.
Another intuitive conclusion about insulation is that completely filling the cavity is better than 1/2 filling. Once again, the laboratory data clarifies things for us. We find that a little insulation does the job and that adding more brings little if any, improvement. See the progression of STC in the data below. Note that adding a mere 2 1/2” of insulation adds 7 STC points to the ceiling. Keep in mind 3 STC points is a very small difference in sound level. A human can’t hear a 2 point STC difference, and most people can’t hear a 3 point STC difference. So if we take the 2 1/2” of insulation and triple it, we only get a 3 point improvement for all that effort. A 3 point improvement that many people can’t even hear.