... This usually means restrictions and permits....
Very few "permits" are ever dreamed up, by any locale, for metal detecting. For example: In the entire state of CA (just using that as an example), I can count on one hand the # of cities (2 or 3?) and counties (1?) that dreamed up such a thing. Thus, no, you can go in asking if they have "permits" in various cities and counties till you're blue in the face, and will usually just get a blank confused stare, in return.
Now as far as "restrictions", it depends on what you mean. If you mean restriction on metal detecting (as a specific), then no, even there you'll find very few at the city and county level (and even state levels, believe it or not), that have something that truly specifically said "no metal detecting". On state park levels, sure, probably something about "cultural heritage" blah blah, but not often that it gets specific enough to say "no metal detecting". And at city levels, rarely any cities that truly had that in those words.
But if you meant "restrictions" with verbage about "defacing", "altering", "digging", "removing", etc..., then sure, if you think those things preclude detecting, then you might as well stay at home. Because I gaurantee you verbage like that exists on every speck of public land. HOWEVER, reasonable people KNOW those refer to the end result. I mean, so long as you leave no trace, then you haven't alter
ED or defac
ED anything. And let's be serious: the "remove" or "harvest" or "take" verbage was never meant to apply to md'rs. Those are so that no numbskull thinks he can take home park benches, cut down tree limbs for fire wood, etc... Hence, no, if there weren't a specific rule that truly said "no metal detecting", then I do not take the other stuff to necessarily apply. Just steer clear (low traffic times) when busy-body who might gripe are present.