mexico
I think Mexico requires a visitor's visa for even green card holders of some countries (after 9/11). Morever, countries in proximity do have special rules. Eg., Australian permanent residents can visit neighboring New Zealand without a visa. However they can not visit Singapore without a Visa, even though Australian's don't need a Visa. More importantly a green card only allows you to live and work permanently in the US. It can always be revoked for certain deportable causes including indulging in prostitution, some convictions including consuming probhibitive substances such as drugs.