SINGAPORE—The trial of a teen video blogger charged with disparaging Christians is showcasing Singapore’s struggle to adapt its tradition of censorship to the realities of the digital era.

Sixteen-year-old Amos Yee is the latest online commentator to be prosecuted in the past year, on charges ranging from contempt of court to sedition. The video for which he was charged was posted on YouTube after the March 23 death of the country’s founding ruler, Lee Kuan Yew, and while it mostly concerned Mr. Lee, it also contained unflattering...