For example forgetting the name of someone you know.
I did that once! I was a member of a Toastmasters Group, and was chairing the meeting. My brain melted down, and I couldn’t remember the name of our Sergeant-At-Arms. GAWD, the embarrassment!
(The lesson to be learned is, when chairing a meeting of that nature, write down the names of those attending!)
There was also the time my tongue slipped, and I said the word “nigger” when I meant the word “trigger.” Holy shit, I wanted to die.
Also, since the OP put it in the subjunctive, I would be embarrassed if anyone saw my personal stash of porn. Nothing illegal, I hasten to add, but stuff that’s – ah – “alternative” in outlook.
Oh, public speaking is fertile ground for embarrassment. Shortly after I arrived in Japan I was giving a little speech about me and how I had come to Japan etc. and I ended it by telling my future English students I was delighted to be here in China. Jeez I wanted to crawl under a desk. The Japanese in the audience, to their credit, made not the slightest reaction to my bungle.
As far as my personal stash of porn goes, I don’t know how embarrassed I would be if someone discovered it. It goes much farther afield than the traditional tab A in slot B, but then it’s porn.
Having a dream where I’m sitting on a bus bench stark naked, and suddenly realizing it’s not a dream.
Aw c’mon, pics or it didn’t happen! 
Leaking any kind of bodily fluid in public is pretty goshdarn embarrassing.
A couple of nights ago I got home after work desperately needing to poop. I made it. I made it by about 5 seconds. I’m not kidding.
I walked in the door, kicked off my shoes and shedded my coat on the way to the bathroom. If I arrived 5 seconds later I would have shit my pants.
I’m temporarily on a prescription medication whose side-effects include diarrhea. If this happened 20 minutes earlier while car-pooling I would have died of embarrassment.
My sphincter was quivering and convulsing like Michael J. Fox in a freezer.
Being that guy who literally misses the bus/train by a second while everyone on said vehicle can see you and are laughing. I know it’s not a biggie, but you do feel small knowing that you could’ve avoided it through better planning.
Walking into your old office when you just got a new office.
Even blood? :dubious:
Menstrual blood, especially.
Yeah, i literally could not imagine a more embarrassing thing to happen to someone. I mean, you just got the new one!
Falling out of my wheelchair in public. While i know that i am perfectly fine and need no help, i also realize that what just happened looks much more dire and serious that it, in reality, really is. So i mentally cringe when this happens in the presence of strangers because i know what is going to quickly follow my fall. Of course, im grateful for such help but its embarrassing as hell, not to mention completely unnecessary.
I hadn’t even though of pooping in public, that would probably do it.
Any of a million things my children have done that make me look like an irresponsible mother (how could a child learn that word?), an abusive mother (do I have the only children that could throw themselves at the ground in a way that makes it look like I pushed them?), a bad mother.
When I was a teenager, I would have been mortified if somehow my crush on someone who didn’t know I existed was revealed. And everyone, including the crush, laughed at the absurdity.
When I was in seventh grade our band sold magazines. The winners of each class in high school won a day trip to the Ice Follies. I was the winner for seventh grade.
It was a little intimidating to be the youngest one going and I didn’t have any older brothers or sisters so I always felt like I was out of the loop as regarded what was “cool.” So mostly I spent the hour-and-a-half ride listening and laughing when appropriate. Couldn’t go wrong there.
As we got near our destination I finally felt at ease enough to contribute so I asked the band instructor, “What time does the Ice Sollies fart?”
Then I spent the rest of the day being quiet again.
I’m very contented to never have to be thirteen again.
How about: knocking someone else out of their wheelchair in public! I was trying to push (it was a friend, and I’m big and strong) but I did something wrong and the chair went right over. (I’m big and strong and damn clumsy.) People looked at me like I was the worst damn monster since Jack the Ripper.
It would be pretty embarrassing for me as a man to deliver a baby. Like, being a man, then doubling over on the sidewalk and out pops a baby. That would be a difficult situation for me.
How about I have the power of flight, but to use it I must take off my shirt and flap my giant nipples around. Honestly if I did that over a city with hella people seeing my big nipples flapping, I would be pretty embarrassed. Or they are small nipples but I am just able to flap them fast enough.
Actual embarrassing moments: Getting poison ivy on my face and it being red and puffy to the point where you can’t open one of your eyes.
Going to give a speech in front of a bunch of the class and someone shouted, “you’re fly is down!”
I’d be embarrassed on they’re behalf. 
I’m more often embarrassed by the actions of other people, rather than my own. Having a friend who went up and tickled the feet of a girl who was sitting on the pavement while on a night out… that was awkward as hell. She started yelling at him, and I was standing around feeling absolutely mortified. He was well on the way to intoxication and didn’t realise what he had done, or even why she was abusing him roundly, so he just walked away, leaving me to make a half-hearted apology on his behalf and follow him. In general, when anyone I’m with is a dick in public, I find myself cringing, mostly because I assume that people will assume I condone or at least find that behaviour acceptable.
I have forgotten names on numerous occasions, and perhaps I should be more ashamed of that than I actually am - but I always know who the people actually are! I do, however, remember being deeply, viscerally ashamed during my teenage years (fertile time of life for awkwardness at the best of times) by a clearly visible erection in my shorts, which I had absolutely no control over and which was the cause of much giggling among the female members (hah!) of my class. Damn thing stuck around for most of the time we were out playing sports. Maybe I should have beaten it with a cricket bat.
Miss Manners has a bunch of saving remarks/ actions for these kind of situations. For instance, the OP could ask pleasantly: “You don’t happen to remember your name, do you?”
Replying to someone who isn’t actually talking to you (for example, they look at you and say ‘Hi’, but they’re speaking to someone directly behind, or they are using a bluetooth headset etc)