Replace drop ceiling with drywall and soundproofing
I've been interested in this topic for years, but 'back in the day' there wasn't much info on the 'net about it.  

Jacks post about RC1 made some sense, so I took a deeper look and found that while on the surface it may seem that he was correct, the use of RC-1 is not recommended by the Steel Stud Manufacturers Association (SSMA) and it's lack of industry standard or specification for its construction is concerning.

To sound proof a room you need to do 4 things:


Decouple The Framing
Install Absorption in the Cavities (Insulate)
Add Mass (Drywall)
Add Damping 


Decouppling

The best sound isolation comes from a double stud wall with green glue. But, installing staggered studs to hang your drywall should be more effective than direct attachment to RC channel.





For a ceiling, you likely don't want the reduction in ceiling height and increase of cost of double studs. Thus a staggered-stud, or sound proofing clips with furring channel is likely your best path forward. According to the SSMA, to use soundproofing clips you should hang 7/8” Drywall Furring Channel aka hat channel (not to be mistaken with RC-2) from the soundproofing clips at a 48" span. RC-1 as well as RC-2 won't fit inside RC clips. 



Insulation

While foam (open or closed cell) is superior for thermal it is distinctly worse for acoustic, just use the cheapest R13 for the walls, and R19 in the ceiling you can find.

Adding mass (Drywall, Plywood, OSB, and Cement Board)

For sound to conduct through a wall, it has to actually move the wall ever so slightly . A heavy wall is harder to move than a lighter wall. Standard 5/8” Type X is best as it is 70+ pounds a board, and cheap at $7 a sheet. Use two layers. Only mud and tape the final layer. Adding more than 2 layers doesn't improve things much, especially in the bass range:


Damping

There are many field applied damping compounds such as green glue that translate the vibrational force laterally converting the sound into heat. If we prevent the drywall from vibrating, the mass, insulation and decoupling become are more effective: