This is not meant to ridicule anyone who makes these mistakes. Often, mistakes like these are made because we were raised by people who made them. If ignorance is the cause, I hope this will help.
"I could care less..." If you think about what you're saying, you'll quickly see that this doesn't make sense. If you could care less, that means you care and the whole point of this phrase is that you don't care. If you "couldn't care less" that means that you don't care at all because you couldn't care less than you do. This phrase is usually said with some passion and you don't want to passionately mis-state your position. Especially in an argument. That would undermine your position and your position might be right.
"Could of," "would of," "should of." This is often the way we pronounce the phrase because of our lazy speech habits, but what we really mean is, "could have," "would have," or "should have." You'll sometimes see these spelled as contractions--"could've, would've and should've." This problem is a written one, so when you make this mistake, you've created a paper trail. That could end up being embarrassing.
Then there's the confusing usage of I and me. How do you know which one to use? Let's look at a couple of examples.
George and me are going to the movies.
Is this correct? No. It should be "George and I are going to the movies." How do you know? Leave George home. Now would you say "Me am going to the movies." or would you say "I am going to the movies." There's your answer. It should be George and I.
Come sit with Alice and I.
Is this correct? No. It should be "Come sit with Alice and me." This is simple. Kick Alice off the couch and what have you got left? Come sit with I? or Come sit with me. Of course it should be me.
Are there any common mistakes you've noticed?