Acting on your instincts
We are intelligent animals, trying our best to pile up tons of rules and morals over our basic instincts. We might have succeeded in dulling them out but they will never disappear completely. There are some moments, when our body acts on its own, or we shout on top of our lungs, it is all part of our survival instinct and you totally shouldn't apologize for being caught in the moment. Besides, having doubts and being slow can not only hurt you but also the people around you, so just go for it, think later.
You want to please the people around you in any way you can, saying a firm No will no doubt upset them so you have problems doing it, or have to apologize profusely when uttering the word. Welcome to the world of betas, where assertiveness is a foreign concept. Saying No and inserting your opinion into the conversation is not something you should be afraid of. Have some self respect and stop trying to be the person others can use on a whim. When did No became a dirty word anyway?
Men crying is totally unmanly. Women crying is unavoidable and annoying. So please, keep all these feelings to yourself, fit your stereotype. Never mind that crying can actually make you feel better afterwards. Apparently crying is healthy, so doing it from time to time without a good reason is perfectly fine. There is no need to feel inferior and if your group mocks you for shedding a tear at something they don't particularly care about - ask them why they think expressing your true feelings is so taboo.
The teachers and your higher ranking colleagues at a new work place have this air around them that makes them look scary. So you don't ask questions and regret it later, or ask them but make sure to apologize first, like you did something wrong, or you are too stupid to figure out everything by yourself. No reason to do that, no really, unless they are running to get out of there, ask them as many questions as you can and pay them back by doing your work better with the newly acquired knowledge.
The things you like
We live in interesting times, where internet decides what is cool and what is cool to hate. Liking Justin Bieber or Twilight for example will not make you a popular guy in some circles, and if you get found out your only way is to apologize for your bad taste.........what? No really, there is a big difference between enjoying something and judging the people, who enjoy something else. One is perfectly harmless, the other can be offensive. If you are a guy who likes something the internet tells them to hate, don't worry about it and don't feel you are doing something wrong.
Not having a "life" / Being a nerd
Another thing, which became popular the moment internet became popular, is how people are being judged based on their interests, and how they spend their free time. Now lets make one thing clear, staying in front of your computer for 12 hours each day and not leaving your house at all can't be considered good. However there is this notion that if you are not meeting up people and partying non-stop you are somehow a no-lifer, who prefers playing computer games/reading books/building robots. In truth the quiet introvert guys, who prefer solitary entertainments, are much less prone to causing trouble, so there is absolutely no need to apologize for making the decision to skip a party and go watch Star Wars for 12th time instead.
Having poor parents/Having rich parents
They are different sides of the same coin and the coin is called "being different" . Sometimes you won't have money for whatever is the popular thing, or maybe you are rich but living in a generally poor suburb and your peers will look at you with jealousy, and distrust. It is not something you have control over now, is it? You can be a nice guy through and through, and still get slack because of your family. Don't apologize, don't care is the right attitude to go for, eventually the others will grow up too.
It is not only a problem in schools, bullying at the work place is a common thing. Or maybe your neighbor is an a-hole, who needs a reality check. Either way you can't let them do their thing, you have to fight back and when the time comes to answer for your actions - don't apologize. Pay them exactly what they deserve and who knows, maybe you'll become friends afterwards instead of hating each other for the rest of your lives.
As long as you take full responsibility for your choices and own up to them you don't need to apologize that you didn't become what your family wanted you to become. Nope, you can't be forced to become a dentist if you don't want to. Never say sorry and never feel bad, especially after you succeed in achieving your dream. Life is at its best when you are doing what you love every day.
The way you look
In some situations there is no choice but to suit up and apply makeup, in most situations your choices of haircut, dressing and eyeliner should not be judged. Wear what you like, smile all the way, don't care about confused looks. Don't say sorry if you are a guy who likes wearing pink, or a girl who enjoys baggy trousers, show to the world you are not bothered by your tastes.