Mostly quiet disc brakes
 

My rear brake has brand new pads, fairly new rotor, and is basically working great. It’s quiet except just when the bike comes to a complete stop. No noise braking and slowing to a stop, but squealing at the very end.

I'd wait till it's not so new and see what it's doing then. Of course you should check everything you did during the install of them to be certain you didn't cause the issue. Same for rotor if it was also recently replaced.

You might bed them in if you haven't done that. However, when done wrong, you'll have more costly resolutions. If you just keep riding it and braking somewhat firmly when you get the chance, then it'll eventually in 150 to 300 miles be bedded in. If you do try to bed them in, then remember not to come to a complete stop. Let the brakes cool by continuing to ride for a time after each application.

You could also try very lightly sanding the pads with some fine sand paper.

But…
 

Why does it only happen and the very end of a stop!

In my experience, it takes a bit for new brakes to bed in. Shimano has a specific protocol for doing that. Basically get up to speed (15-20 MPH), apply one brake (front or rear) not both. Use moderate to heavy brake pressure to slow down. But don't stop. Get down to a walking speed. Repeat that ~20 times. Do it again for the other brake. You can certainly alternate between them.

I think the brake dust itself is a factor in having quiet brakes. And I'm sure all of this depends on the pad brand and type.

Having said all of that, very rarely (like once every 3-4 months) for no reason I know of, I'll get a bit of brake squeak just as I'm stopping. Probably just the right combination of speed, pad condition, temperatures, and brake pressure.

When I change my waxed chain, monthly, for a freshly waxed chain I also remove my disk pads and give them an inspection and light sanding. I also clean the rotors with rubbing alcohol as well as inspecting them for wear, etc. My brakes don't squeak or squeal...unless I ride on wet roads then all bets are off lol.

I dunno - I don't do much of anything to my 3 disc brake bikes. Road, CX, MTB - squealing isn't really an issue. Wet conditions will cause noise, but all the other times they are quiet.

No special cleaning... about once a year or so I may sand the pads and rotors down a bit. Nor do I really "bed" them in properly. I do hear other bikes that sound like an angry goose - not sure how or why.