welcome to the forum. Your concerns are fairly typical (and understandable) : A newbie gets a detector. They hop on internet forums hoping to learn all-that-they-can, before striking out. Eh ? But lo & behold they see hints of "allowed" and "not allowed". And/or talk of "permits" at certain places. Or talk of someone having been hassled, blah blah blah.
Thus they worry "gee, is it allowed where I'm at ??".
Here's the true skinny: It's like shark attacks. Extremely rare to be attacked by a shark. Even though thousands and thousands swim in the ocean each day. But the moment someone gets bit, guess what makes front page news ? And then guess what everyone fears when they go swimming next ?
So too is it with md'ing public places: If you avoid obvious historic sensitive landmarks, and perhaps a scarce few types fed, you have nothing to worry about. Just avoid NPS for example. And anything with a historic monument theme.
Also I notice you're asking where it's "allowed". That's sort of the wrong approach to begin with. You need to ask where (if any) that it's NOT allowed. Because you're never going to find express allowances (eg.: a sign or rule that says "metal detecting allowed here"). Instead you can assume that if it's not expressly DIS-allowed, that ........ presto, it's not dis-allowed.
Of course you need to bear in mind that you're in an odd-ball hobby that draws lookie-lous and has connotations. So, like nose-picking, you need to be a little discreet at the times you (especially at nicely turfed parks) Eg.: don't go waltzing over people's beach blankets and through ball-games, avoid archie conventions times, etc...