Singing is a skill introduced in The Sims 4 with the City Living Expansion Pack. It's one of the least practical skills in the game and has little utility outside of minor socialization, but it can be used to help bolster relationships.
Singing can be started any time with any Sim by executing self-command called "Practice Singing." Like the other musical skills, singing with a low Singing level around other people tends to create embarrassment for the singer and other negative emotions for the listeners. However, as Singing goes up, listeners will gain buffs.
Singing mainly comes into play while out on the town at Karaoke bars. Listeners will be affected by the singer's skill level, for better or worse, along with any booing or cheering as a result. Singing does not impact any career tracks; the Entertainer career track was not retroactively changed to include being a singer.
When Sims sing in the shower, it does not raise their Singing skill level. However, their Singing skill level will impact the quality of their songs for your own ears. The quality of shower songs has no impact on anyone around them