As a Wikipedian, by far the most common edits I have made are fixes to incorrect punctuation. Here are the mistakes I see most often. This list is not comprehensive because it is restricted to the errors I encounter most frequently. I have also tried to restrict the list to things that are universally regarded as incorrect, as opposed to stylistic issues like the Oxford comma.

Some examples were taken from Wikipedia articles, whether they're actual errors I found or they were used in an article as a sentence illustrating an error. Others were made up on the spot.

Commas

Here is a fundamental rule of comma usage that is often neglected:

When you have an opening comma, you must also have a closing comma. The only exception is when the closing comma would come after the last word in the sentence.

Commas are not necessary:

Avoid comma splices: do not use commas where it is more appropriate to use a semicolon, colon, or dash. Sometimes it's best to add a conjunction after the comma or split the sentence into two sentences.

(The article on comma splices has a few more variations of this sentence, but this should be enough to illustrate the idea.)

Parentheses

If you wouldn't use the punctuation without the parenthesized phrase, then there shouldn't be any around the parentheses.

A sentence that begins outside parentheses ends outside parentheses.

A sentence that begins inside parentheses ends inside parentheses.

What do you do if you want to have an entire sentence inside parentheses that itself exists in another sentence? In such a case, you probably should rephrase to avoid the problem.

Quotation marks

Always nest quotation marks correctly: single quotes within double quotes and vice versa.

Do not put a commas after a quotation that ends with a period. Replace the period with a comma.

Note how the comma goes inside the quotation marks. When the comma stands in for a period, this rule applies even if you're using the so-called "logical" or "British" style.

Finally, there is no need to put quotation marks around a passage that is blockquoted.

Apostrophes

Semicolons

Semicolons are often used where commas or colons would be more appropriate (but be sure to avoid comma splices).

This is a sentence pulled from an actual Wikipedia article (albeit trimmed for brevity). Its author might have been thinking the semicolon is necessary since commas were used to delimit the items of the list, but it's not, since the word "and" clearly signals the last item of the list.

Miscellaneous