It depends on how you got onto the list in the first place.
Sometimes “unsubscribe” is exactly the right thing to do.
Sometimes you want to avoid it at all costs.
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If you subscribed to the list
If you actually subscribed to the email list yourself — by filling out a form, or joining an organization that put you on it, or by some other explicit step you can recall — clicking the “unsubscribe” link is exactly the right thing to do. 99% of the time you’ll be removed from the mailing list without any hassle.
Naturally, there’s that other 1% where things don’t go as promised.
I give lists a couple of days to remove me. Some lists tell you “It may take up to 48 hours for you to be removed.” There should be no need — unsubscribing should be instantaneous — but I’ve seen lists claim it’ll take up to a week.
I may not give them a week, but I’ll usually give them a couple of days.
If you continue to get emails after that, it’s time to call it what it has become: spam.
If you didn’t subscribe
If you’ve been placed on a list you didn’t ask to be on, then by definition it’s spam.
NEVER click an “unsubscribe” link in spam. As you’ve seen, you’ll just get more spam.
Clicking on the unsubscribe link in spam tells spammers that you actually read email sent to this email address, and thus you become a bigger target.
The best way to deal with spam is simple: if it appears in your inbox, mark it as spam every time it arrives. Depending on your email program, eventually it will learn that email like this is spam, and should either be rejected outright or delivered into your spam folder, where you need not look at it.
One “this is spam” caveat
If you subscribed, try to unsubscribe first.
“This is spam” should not be your first choice when you want to stop receiving something you asked for. Give the provider the opportunity to handle things the proper way. Only when they fail to remove you should you then mark it as spam.
The issue is simply that when you mark something as spam, it makes it less likely that others who actually want the mail will get it. Spam detection is complex, but one of the signals used is that “this is spam” button. If enough people hit it, the mail will be more likely to be filtered as spam for everyone.
Clicking “this is spam” on legitimate email unjustly punishes the folks who play by the rules.