I'm applying for a visa for Thailand with a US Travel Document I-327 (I'm a Permanent US Resident). Should I put "STATELESS" in the nationality box? My nationality says Stateless on my Travel Document. I'm not sure if I should put down what's on my document or what my actual nationality is (where I was born).
Your current nationality is stateless, so that is what you would put there.
Your nationality at birth would be of the country you were born in that no longer exists as it was before the breakup of the Soviet Union.