On the class of noetherian rings, the property "having finite Krull dimension" holds for every local ring, hence is equivalent for $A_m$ at all $m$ maximal, or for $A_p$ at all $p$. However the property is not local since there are noetherian rings of infinite Krull dimension (Nagata).

If you want the property to be defined over all commutative rings, just build-in noetherianity by changing P into: "is non-noetherian or of finite Krull dimension". ;-)

As to the strongest such property, it is probably P="being local". Indeed, it is a non-local property but it holds for all $A_m$, or equivalently for all $A_p$. At this stage I'm wondering whether I understood the question right. :)))

On the class of noetherian rings, the property "having finite Krull dimension" holds for every local ring, hence is equivalent for $A_m$ at all $m$ maximal, or for $A_p$ at all $p$. However the property is not local since there are noetherian rings of infinite Krull dimension (Nagata).

If you want the property to be defined over all commutative rings, just build-in noetherianity by changing P into: "is non-noetherian or of finite Krull dimension". ;-)

As to the final such property, it is probably P="being local". Indeed, it is a non-local property but it holds for all $A_m$, or equivalently for all $A_p$. At this stage I'm wondering whether I understood the question right. :)))