Hello!

This question is one that seemed obvious to me, but many local judges have ruled the other way - I searched around and couldn't find an answer to this specific question, so here goes!

Albert controls a Polukranos, World Eater that isn't monstrous. Norbert controls no permanents. Can Albert activate Polukranos' monstrosity ability? If so, will counters go on Polukranos?

Thanks!

Yes, on both counts. It's perfectly legal to activate Polukranos's monstrosity ability no matter how many (or few) permanents your opponent controls, and as Polukranos wasn't already monstrous the ability will put +1/+1 counters on it and make it monstrous.

If your opponent happens to control no creatures at the time Polukranos becomes monstrous and triggers the second ability that trigger will be removed from the stack for having no legal targets to choose from, but that won't affect Polukranos or its counters in any way.