$10 notes are still very common here in Australia; which is interesting since the ATMs only seem to spit out $20s and $50s.
I noticed this in the US when I was last there, too… I’d pay for something worth $3.50 with a $20 note (the smallest in my wallet) and instead of getting a $10, a $5, a $1, and two quarters as change, I’d get three $5 notes, a $1 note, and two quarters back as change.
It’s an interesting topic, at any rate!
Where are these ATMs that “give out” money? All of the ones I know of insist that I have money on deposit before they’ll let me withdraw some of it out! 
This question has been asked before (now where’s that search…)
I work in a retail-type place (not a bar) and I see $10 bills all the time. I also get tips along with money that I don’t turn in until the end of my shift. When possible, I keep any tens I get. I like the ten, especially that neat peach color. Right now I have 7 tens in my wallet, more than any other denomination.
One denomination that’s changed in frequency is the $50. Recently I saw one of the old fifties, the one with the small picture of Grant in the oval. I almost didn’t recognize it. That old one circulated so little that hardly anyone ever got one. With inflation, the new one is becoming quite a common note, although no where near as common as the 10.
I hold a bunch of $10 bills. I dislike handling money (most of my transactions are via credit card), but I still need to hold cash in the wallet for emergencies. I prefer clean, crisp, uncirculated bills, so I try to walk around without grubby used bills or coins.
When I go to the bank to request clean, crisp, uncirculated bills, the $10s are the best choice precisely because they don’t get used much.
Do you get your butler to iron your paper money for you in the morning as well?