Just how many outlets do you need for your stereo?

It is a question that cannot possibly be answered. It is dependant on what you own and what you intend to own. You could need anywhere from 4 to 20....who knows?? What is most important is not plugging the power strips/conditioners into different circuit legs. Keep them all plugged into one circuit, preferably a 20 amp dedicated circuit. The niagara 1200 is a great device. Also, the much cheaper furman pst-8. It is designed by the same guy. These two mentioned will give you I believe 15 outlets.
I use both plugged into a single AQ Edison receptacle wired with 12/2 romex (50 ft) to the service panel using a separate 20 amp breaker. Below are a list of what I have plugged into the two...

Integrated amp
Sacd player
Cd transport
Dac
Tuner
Turntable power supply
Phono preamp power supply
Headphone amp power supply
Cassette deck
Digital streamer
Total: 10 items currently...will be introducing my kenwood minidisc recorder and second turntable at some point, so that will be 12 outlets used.....

One circuit may be the optimal way to go, but some who have really powerful amplifiers may want more power than 15 (or even 20) amps for the whole system.  
Also, some don't care for conditioners/filters and would rather plug their gear directly into the wall, particularly analog gear.  If installing new power, having multiple lines should not be a problem if they are dedicated (i.e., directly wired to a single outlet so they do not power anything else) and all run from the same side of the panel.   I believe there may be other best practices such as keeping audio circuits away from circuits powering noisy fans and resistive/variable type devices.  Another option is to establish a sub-panel (like 50 amps) and then run all the audio system outlets from that.  I am fortunate that my 3, dedicated, 20A lines are absolutely quiet.  

@mitch2 @audioguy85  Thanks for the info, it was very helpful. 

The two units from Amazon arrive today - these are the two I mentioned before .  For $250, I am not expecting much in the filtering but the sequencing will be much appreciated.  I will report back on the innards and performance.  Hopefully it will serve as a stop gap for now until I find something suitable for long term use.

Keeping it at one dedicated receptacle avoids a ground loop issue. I suppose if you need more than 20 amps (I doubt it) you could always go with 10/2 cabling and a 30 amp breaker and 30 amp rated receptacle. To me it sounds like over kill.

The two Pyle PS1000 units arrived today.   The PS stands for Power Sequencer and it will sequence 8 outlets  Overall, it is a clean build and well worth the $125 price for each.   The sequencer can be daisy chained so up to 100 outlets can be sequenced, although each unit requires a connection to the AC mains.   See it  here - https://smile.amazon.com/Outlet-Power-Sequencer-Conditioner-Controller/dp/B07P76C2PF/ref=sr_1_6?crid...

The sequencer, voltmeter, temperature gauge, and relays are all controlled by a microprocessor.  Looking at the manual, I see a 24 dBm / octave filter on each outlet.  The filter is made up of two X caps and two common chokes so there are two 12 dBm / octave filters back to back on each outlet, making up the 24 dBm filter.  No values are specified, so I can't sim it in Cadence.

That is the good news.  They somewhat bad news is they also have the typical Y capacitors from line and neutral to ground,  Although the Y caps can improve noise filtering somewhat, it also bleeds some line current onto the grounds, so it may induce a low level hum into the system.   In general, I dislike inducing any current onto Earth ground, but it is a very common practice.  [UL codes are pretty disgusting in this respect, as they allow multiple milliamps of bleed current.]   The primary difference between regular line filters, like this one, and hospital grade line filters, are the lack of Y caps in medical grade filters.
I'll do some more involved listening tests this weekend and next weekend, if I get some time in the lab, I will see how much of that microprocessor clock bleeds onto the outlets and what the filter bandwidths are like.   Hopefully, it is better than the APC unit I have now.