When you really really want to emphasize that no one cares about something. Not only does it feel satisfying to say, it's a quick and easy way to shut someone down when they say something stupid or trivial.
Guy 1: My favorite color is white.
Guy 2: No one gives a shit.
Used as a sarcastic comment for someone who stated the obvious, using a reference to the famousfiction detective Sherlock Holmes, it can be understood as "No way, you are Sherlock Holmes!" which can actually mean "Of course, you dumbass!".
The phrase "no shit Conan" refers to the phrase "no shit Sherlock", but instead of Sherlock it's Conan, the main character from the anime Detective Conan.
This phrase is the weeb version of the "no shit Sherlock".
Normie: Did you know that Conan likes Sherlock Holmes?
Weeb: No shit Conan.