I'm managing another phpBB forum, and there's a very serious problem with spam. It's to the point that we can't even allow account creation because spammers will come in too fast.

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That tricky no/spam popup probably catches most of 'em.

With PhpBB, you can set the number of first time posts (like 3 or so) that require moderation for new forum members posts to appear. Then if they're spammers, their spam posts don't appear on the actual forum, and you can ban the user AND their IP. Once they post 3 legit comments, then you're reasonable sure they're not spammers.

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Either provide enough details for people to help, or sit back and listen to the crickets chirp.
Security thru obscurity is a myth - and really really annoying.

What I did (found it somewhere in phpbb's own community I think) was add a custom profile field with a required specific answer. In my case, it's a number, so only the correct answer is allowed. I even put the answer in text (ie seven when the answer is 7) within the field description, so if a person doesn't know the answer, it's right there. That immediately cut my spam registrations to almost nothing.

On top of that, I've always done account activation by admin only, so the few spammers that do slip through the question I usually catch on that. May not be reasonable for your board though if you get a lot of legit registrations.