Hi there,
I have received complimentary upgrades twice in the past year. First one when I was flying Liberia-Chicago on UA metal, I was only *S back then. I asked politely at the Premier checkin desk if there was an economy plus seat available on the flight (I know from experience that, although it is not a published benefit, if you are *S or *G you may ask for free for an E+ seat at checkin or at the gate, depending on your luck, worked for me 8/10 times). So when I asked for an E+ seat, based on my silver status, she said "oh I see that first is empty, I will see what I can do, in the meantime here are your E+ boarding passes". Then surely they called my name once seated in the plane and we were bumped in first cabin first row. Y cabin was half empty.
Another time was on LX from GVA to LHR, I was traveling with my sister and the girl at checkin asked if she could take my aisle seat because someone with a broken leg was waiting for an aisle seat to become free. I told her there was no problem and we were reallocated other seats. She handed our new boarding passes and said that based on my gold status she was able to upgrade us to C. I agree that in this case it was more of an Op-Up although she could just have given me the former seat from the injured guy.
I would say it happens with a bit of luck, but from my experience it can happen a lot more when you fly on American companies than on European ones. At least I almost always got E+ as a *S when asking at the gate directly, and once complimentary first (not bad out of the dozen flights I flew on UA metal when I was on a short contract in the US).
The old trick of "be nice, polite and dressed properly" may work in the US when you behave normally and do not act like a dog standing there and asking for a bone

. I believe asking at the gate has a greater chance of success than asking at checkin. Although you should never expect anything else than what you paid for, sometimes nice surprises happen. Also, I was working for a foreign embassy at that time and maybe being a foreigner with a strange accent helped... Maybe not... Who knows
My 2 cents!