Firstly, most of us do not bother if another stutters, inserts an "um" mid-sentence, or fidgets a little. Yet for intelligent people like us, we know that a little slip-up could cost dearly. I'm going to focus mainly on public speaking as I think that's where most social blunders take place. Here are a few tips to get yourself in the best shape:
1) Practice.
Practice makes perfect. We only perform at the average level of all of our abilities, so if you want to stand out, you need to polish yourself and pull up your "average" performance.
Some practical ideas on how to do it:
a) Public speaking: If you're terrified of crowds, get your friends and/or family together and speak to them. Every day. Without fail. Make sure you incorporate their feedback in subsequent training sessions, because as someone rightly pointed out, "perfect" practice makes perfect. Repeating the same mistakes merely reinforces those mistakes.
b) Writing: Master the art of writing relentlessly. Read and write every day -- a writer once told me to "read 2k, write 2k", 2k meaning 2,000 words. It was valuable advice. Don't forget to proofread.
c) Networking: Be good to everyone, especially the people you come into contact with, and be nice to people regardless of how the treat you. A bus operator called Linda Wilson-Allen put her heart into her service, and although she had her share of not-so-friendly passengers, her kindness became known to all. When you are used to treating the people who are the least reckoned kindly, you will treat everyone kindly.
2) Be early. Don't rush.
Being early gives you the benefit of having more time to prepare and calm down. Sometimes questions about the content you'll be presenting may be problematic and you won't know it until you've re-read your stuff. Sometimes you may find a glaring error or omission in your files that you've never spotted before. These problems can be addressed more effectively if we are early and can take several more deep breaths.
I kicked off the open-mic session at TEDxCUHK earlier this month, and as I went over my script, I realised that I made a serious error in mathematics: π was the ratio of the circumference to the diameter, not the radius, of a circle, and I didn't see it at all. (My speech is essentially the same as "Maths Is Useful. How?") It was at that moment at the TEDxCUHK venue that I corrected my would-be blunder. Alas, an urgent message interrupted my thoughts. Since I have been used to wearing a ponytail every day before the event and I wanted to have my long tresses photographed on the big day, it was right after that message that I had to stand up and speak. I forgot to let my hair down literally... That's why the little things you do every day matter.
3) Double-check.
Whatever you do, a second look may show you something you've missed. It often happens in doing mathematics, where a careless mistake ruins the rest of your calculations and hard work, and you end up with the dreaded wrong answer.
Sometimes the way you wish to present yourself looks hideous in retrospect. Last year I was invited to my old school to share about my experience writing about mathematics, and I put on jeans and a T-shirt. However, the other speakers were male and I wasn't too aware of that, and I did not seem to stand out among them as the only woman. I thus am resolved to wear a skirt or dress in upcoming formal presentations, and the more I saw fellow women mathematicians wearing trousers to their presentations, the more convinced am I to require that I maintain the distinction myself so as not to give the impression that doing maths meant neglecting one's femininity.
If there's anything to take to heart, it's that we must allow for the possibility that our best intentions could be wrong. Therefore we ought to be humble enough to learn from our social blunders before moving on.