I've tested two of those. It's hit and miss. If it's not closed perfectly flat (and it's metal-lined fabric, it's not always perfectly flat even if you're careful), it'll receive phone calls just fine.

A faraday cage works for incoming signals, but not so much for outgoing. It also depends on the wavelength vs size of mesh.

I think using a solid cage is the best bet.

Hi 'm463, you seem knowledgeable about this subject, so I have a few questions. Could you please answer the questions below or direct me to where I can learn more?

1. If a phone is off e.g. iOS’s General->Shut Down, then can it still receive and transmit signals?

2. What is the best kind of cheap case / enclosure for a cellphone that would prevent signals from being transmitted or received? Can I just wrap a cellphone in aluminum foil and place said wrapped cellphone in a Tupperware / plastic sandwich container?

3. What is an effective way for an RF layperson like myself to detect whether or not my phone is transmitting or receiving signals while it appears to be off e.g. RF tool or measurement device?

I just want a way to know and be completely certain that “off” means “off”.

Those are good questions and I'm not an expert. I was a part of a discussion once where someone mentioned that faraday cages mostly work for signals entering.

However:

1a) You don't know (because your phone can pretend to be off). You need to remove the battery (and also remove hidden batteries)

1b) some phones support NFC, which can theoretically be used when the phone is off.

2) I suggest being familiar with: https://en.wikipedia.org/wiki/Faraday_cage and your cellphone before making a decision

3) I don't know, but this would help with #2