For the best results, pay attention to both the floor above and ceiling below. (Insulation between the floors is of little value.) See the link below. Also, a big source of sound transfer are flanking sounds that come through such things as the drain pipes and the holes surrounding them and supply pipes and electrical wires and conduits.

Lead drains will greatly dampen sound--and drain your pocketbook. Instead, build drywall boxes around drains--a couple layers thick if you have the space--and stuff the space around with sound-deadening insulation. Even fiberglass wrapped tightly around the pipes will make a tremendous difference.

Here is a link that might be useful: Download Background on Floor/Ceiling Noise Issues

The best floor sound separation detail I have used is resilient channels that support 5/8" or double layer 1/2" gypsum board at the ceiling.

Flooring treatment can be homasote or other specialty acoustic layers or sleepers with another subfloor and then carpet with pad or gypcrete underlayment, etc.

Waste pipes that travel through different apartments should be no-hub cast iron behind a double layer of drywall. There are systems that can wrap PVC pipes (see link below) but they require a lot of labor to install. If the run is short cast iron is best IMHO.

Here is a link that might be useful: encapusated PVC pipes

I watch a show called Income Property on cable. I have seen them use a special batting insulation and drywall in between floors. That made a firewall and help with soundproofing.

The only difference between that detail and a typical residential floor-ceiling assembly is that the drywall in the fire-rated assembly is designed to stay in one piece after exposure to fire but it would provide no increase in sound separation.

Air borne sound abatement detailing generally uses mass, resiliency, and absorption. In a residential ceiling the mass is provided by gypsum drywall, the resiliency is provided by resilient channels and the absorption is provided by thermal or acoustic insulation.

Structure borne sound abatement usually consists of resilient coverings or light weight concrete toppings on the floor above.

Ceilings should be easier than walls because there is plenty of room for insulation, less perimeter joints, and it is easy to hang a ceiling resiliently but then there is the foot traffic above to contend with.

At walls the challenge is to provide reasonable sound separation without making the wall too thick which leads to many products from resilient layers in drywall to Green Glue between layers.

The kind of insulation doesn't usually matter unless it is sprayed which seals cracks very well. Cracks create major sound leaks in field build enclosures.

Lead drains

Meant cast iron stacks.

Multiple attacks on sound transmission may yield better results than relying on one approach.

I've used resilient channels numerous times. But since just one misplaced screw can shortciruit the whole system, I'm more inclined to use sound-damping drywall instead where the budget permits.