According to immigration attorneys, U.S. Citizenship and Immigration Services (USCIS) has been sending out Requests for Evidence (RFEs) in cases where applicants for Adjustment of Status (a green card) fail to fill out Part 3, Question C of Form I-485. An RFE asks the applicant to correct the problem and submit additional materials by mail before USCIS moves the case forward -- which, obviously, can create delays of many weeks.
This question on Form I-485 simply asks about whether you are or have been a member of or affiliated with any organization, club, foundation, party, society, or similar group, as well whether you have performed military service, and to provide dates and other details for each one.
Although the question also instructs applicants to write “None” if they haven't been a member of any such groups, some people overlook this instruction, assuming that USCIS will see from the blank spaces that they've had no such involvements.
Or perhaps some applicants are actually worried that group involvement is a negative thing. True, if you have been a member of a group that the U.S. believes to be associated with terrorism, you may be found inadmissible to the U.S. (ineligible for a green card or other entry), and should absolutely consult with an attorney before submitting an application for U.S. lawful permanent residence. See "Terrorism and Other Security Violations That Make U.S. Visa or Green Card Applicants Inadmissible" for more on this.
But not all group affiliations are negative -- quite the opposite. If and when you apply for U.S. citizenship (naturalization), you will be asked a similar question (on Form N-400, Part 10, Question B.). Membership in some types of groups, such as charitable or community activities, will be looked upon as a sign that you have the "good moral character" required for U.S. citizenship. But if you claim a group membership that stretches back to before you applied for the green card, and you didn't mention it on the Form I-485, the citizenship examiner is going to wonder whether you were lying then or are lying now.
The upshot is that, in filling out Form I-485, you will need to take particular care to accurately complete this section describing your group memberships or state “None” if that is the case. Again, if you're worried that one of your memberships will hurt your application for a green card, consult with an attorney.
Be similarly careful to leave nothing blank on other USCIS forms. If the answer is "None," state none. If the question doesn't apply to your situation at all, enter "N/A" for "Not applicable."