I tried yet another USB port (which is not one of the blue ones) and was able to go more than 27 minutes without a dropout, before I had to go.  Later, I tested for about an hour in Sonar, and didn't have any dropouts, just the normal crackles whenever I try to move clips around in real time (which is what I've always gotten with Platinum regardless of sound card).
 
So it could be that I've picked a "lucky" USB port.  I will test more tomorrow to see if I get any dropouts.
 
If I do, then I will try to dig up a USB PCI card from one of the boxes of old hardware in my closet.  I'm pretty sure I have one laying around.  I never gave those things much consideration until now!
 
I ran a couple of latency monitors a few times and didn't see anything alarming.  That's not surprising, since I wasn't having dropouts with my other sound cards, and the latency monitors don't seem to be checking specific audio devices.
 
As for startup items, I'm pretty good at periodically going through msconfig and services.msc and pruning out the optional stuff.  And I avoid installing a lot of apps.  Mostly just stuff for work, games, and multimedia.  My antivirus is AVG Free and it hasn't given my audio-related problems in the past, so I don't suspect it of causing problems now.
 
But I am really unfamiliar with USB sound cards, so can't really trust my instincts too much!
 
Anyway I'll see how things go tomorrow on the potentially "lucky" USB port.  Fingers crossed (for the Nth time this week...)!

My antivirus is AVG Free and it hasn't given my audio-related problems in the past, so I don't suspect it of causing problems now.
 
BINGO! 
Even though you don't think it's causing issues this is a highly suspect culprit. Most report that only Windows defender seems unobtrusive but most Anti Viruses do interfere.
 
You should try disabling all known sources of trouble and slowly turn them on.   
I don't think a PCI USB port is your solution. 
Unless you have a zillion things hooked up to all your ports that are sucking power from the Scarlett. I think it's a matter of choosing a port that is free and clear of other activity. My interface is sharing the back ports with my Stienberg dongle, and  extension cables used for external drives. 
The other hub has my Roland controller, which runs on USB power, my digital drums and my mouse/keyboard. 
My interface uses wall power, in your case if you have 2 devices that both need USB power on the same hub your going to have a problem. 

Cactus Music
My antivirus is AVG Free and it hasn't given my audio-related problems in the past, so I don't suspect it of causing problems now.
 
BINGO! 

I thought the same...I used AVG Free for years on XP but it caused problems with W10, had to uninstall it.

What do you use for AntiVirus instead?

I've been listening to the Scarlett for 3 hours straight this morning, and haven't had any dropouts. So, maybe the last USB port I picked to try is indeed "lucky".  I have no idea why, though.  Hmm...
 
Anyway, I'll keep using the Scarlett today and see if it gives me any dropouts.
 
Does the fact that one USB port seems to behave better than others suggest that the problem is not AVG related?

Cactus Music
...
I don't think a PCI USB port is your solution. 
Unless you have a zillion things hooked up to all your ports that are sucking power from the Scarlett. I think it's a matter of choosing a port that is free and clear of other activity. My interface is sharing the back ports with my Stienberg dongle, and  extension cables used for external drives. 

+1
 
I'm the one who suggested the PCI card -- with the condition that he try all of the other ports first. IOW, make it a last resort!  Because, in fairness, the PCI card would solve the problem, it's just that he probably doesn't need it since one of the other USB2 ports would likely work fine.
 
Edit:  OP's last response confirmed it -- He was using a USB3 port!  Moved to a USB2 port and it seems to be okay.

Tripecac
 
Does the fact that one USB port seems to behave better than others suggest that the problem is not AVG related?

Yes, if you're no longer detecting any other interruptions, the problems were likely due to the USB3 port.

Tripecac
What do you use for AntiVirus instead?

Windows Defender

I don't think the port you picked is "lucky."  I think its actually a USB2 port.  If the original port was "blue" it is a USB3 port and some USB2 interfaces don't like certain (any?) USB3 ports with Win7.

Oh, and Windows Defender is not what you use with W7 , You use Microsoft Security Essentials 
You can install Windows Defender but it only does the Firewall or something basic. In Windows 8 and 10 it's a full on anti virus,, but not 7. 
I find MSE very buggy for the updating these days,, it stalls