It was always after placing a letter, after a change in control (stopper or missolve), or at the beginning of the word when a contestant could buzz in. They were lenient in the time (for a time, a quick honk-honk was just a prompt to draw letters -- I believe they would allow a solve soon after it) but when you dropped the letter tile, the solve period was closed.
And, in speedword, you have to wait until Chuck says go.