When can I spy?
Spying is a World War function, and is unlocked as you advance through Harriet Tubman’s skill tree in the University.  Once you have advanced her tree far enough, and developed the Spy technologies, you will be able to perform spy actions in World War!

Can I choose which target I spy on?
You are restricted to your opposite number in War.  So, if you’re listed as number 7 in your Alliance, you can spy on number 7 in the opponents Alliance.

Can I pick which buildings to spy on?
There are three skills Harriet Tubman has for Spying, each of which corresponds to one Building in your enemy’s Village.  You can spy on an enemy’s University, Library, and Museum, once you’ve unlocked their respective skills.  This allows you to see the opponent’s University Leader development, Library Research, and active War Wing Museum Artifacts.