I have a green card, have been in the U.S. for six years, and speak English pretty well. However, I am on probation for a crime. Should I apply for citizenship, or will it be denied?
You will need to wait before applying to naturalize – and had best consult with an immigration lawyer in the meantime. A person who, after being convicted of a crime, is placed on probation or parole, or has a suspended sentence, must successfully complete it in order to be approved for U.S. citizenship. (This comes from the U.S. Citizenship and Immigration Services (USCIS) regulations at 8 C.F.R. § 316.10(c)(1).)
There are no exceptions to this rule. Your citizenship application will simply not be approved while you are on probation or parole—no matter how minor the crime. USCIS will either postpone a decision on your application until your probation or parole is completed or ask you to reapply later.
Even after you have completed your probation, however, your troubles may not be over. You will need to look into whether the crime that you committed could result in the U.S. government taking away your green card, as covered in Nolo's article, "Crimes That Will Make an Immigrant Deportable."
And even if you dodge that bullet, you will need to find out whether the crime bars you from receiving U.S. citizenship, or might cause the USCIS examiner to deny you based on lack of good moral character, as described in "Crimes That Will Prevent You From Receiving U.S. Citizenship."