I do not know about the Caribbean or Mexico but you can go to Canada without a passport or a travel document. You only need a green card.
And Canada is a very fun vacation spot.
Is it possible that they're experiencing a backlog because people are renewing their RTD and RP just to beat the fee increase?
I just send my application for RP this week. I hope they'll get better next year after they approved all the people that rushed to beat the fee increase.
Oh well. They approved my wife I-485 in 3 months, mine in 11 months. So I'm not complaining ..... yet!!!!!
Yes you can go to Singapore w/o Visa on RP. It said so in their official tourism webpage.
http://www.visitsingapore.com/publish/stbportal/en/home/about_singapore/US_Homepage.html
American passport holders, US Refugee Travel Document (Form I-571) and US Re-Entry Permit (Form I-327) holders do not require a visa for up to 30 days. Canadian passport holders do not require a visa for up to 30 days.