Writing "woulda" or "would of" instead of "would have."
People both saying and writing "mines." Example: "That one's yours, but this one's mines!" It should just be "mine."
Another common one is confusing "your" and "you're." "Your" is possessive and "you're" is the contraction of "you are."
Some people still insist on saying "a umbrella" instead of "an umbrella."
Some people also refuse to capitalize the pronoun "I." It should be "why would I want to go to the store?" not "why would i want 2 go 2 the store?"
Spelling phonetically. Native English speakers tend to contract words a lot, and if they speak more than they read, they tend to make a lot of mistakes. For example, "probly" and "rediculous" instead of "probably" and "ridiculous." (EDIT: Peter Mortensen put a link in the comments with some good examples. I'll point out, though, that "extant" is also a word. Here's the link: http://www.irosetta.com/questions/433/vowel-substitutions )
Also, learn the exceptions to the "i before e" rule. A few common exceptions are weird, seize, all of the -cies plurals, and caffeine.
People ignore the subjunctive case a lot. Some even believe it doesn't exist in English. It should be "if I were going,..." not "if I was going,..."
If I think of more, I'll edit...